OF 
R.    Tracy  Crawford 


A8TRUHUMI 


THE   SECULAR   VARIATIONS 


OF    THE 


ELEMENTS   OF  THE   CEBITS 


OF    THE 


FOUR  INNER  PLANETS 


COMPUTED   FOR  THE   EPOCH   1850.0,  G.  M.  T 


BY 

ERIC   DOOLITTLE 
«\ 

UNIVERSITY  OF  PENNSYLVANIA 


Extracted  from  THE  TRANSACTIONS  OF  THE  AMERICAN  PHILOSOPHICAL  SOCIETY, 

N.S.,  Vol.  XXII,  Part  2 


PHILADELPHIA 
1912 


To  MY  FATHER, 
PROFESSOR  CHARLES  L.   DOOLITTLE, 

THIS   WORK 

is  INSCRIBED. 


ASTROUOVY 


[Extracted  from  the  TRANSACTIONS  OP  THE  AMERICAN  PHILOSOPHICAL  SOCIETY,  N.  S.,  Vol.  XXII.,  Part  2.] 


THE  SECULAR  VARIATIONS  OF  THE  ELEMENTS  OF  THE  ORBITS  OF  THE  FOUR 
INNER  PLANETS  COMPUTED  FOR  THE  EPOCH   1850.Q  G.  M.  T. 

BY  ERIC  DOOLITTLE. 

(Read  March  1,  1912.) 


TABLE  OF  CONTENTS. 

THE  THEORY. 

1.  Introduction 39 

2.  The  method  of  GAUSS 40 

3.  HILL'S  first  modification  of  GAUSS'S  method 42 

4.  HILL'S  second  modification.     The  work  of  CALLANDREAU  and  INNES 47 

5.  The  method  of  HALPHEN  and  its  modifications  by  ARNDT  and  INNES 49 

THE  COMPUTATION. 

6.  The  elements  of  the  orbits  and  the  adopted  masses 52 

7.  The  formulas  employed  in  the  computation 53 

8.  The  values  of  the  preliminary  constants 56 

9.  The  radii  vectores  and  the  true  anomalies 59 

10.  The  separate  results: 

Mercury  by  Venus 61         Earth  by  Mercury 123 

"  Earth 65            "       "  Venus 127 

"  Mars r." .  .     70            "       "  Mars 132 

"  Jupiter 77            "       "  Jupiter 138 

"  Saturn 82            "       "  Saturn 142 

"  Uranus 86            "       "  Uranus 146 

"         "  Neptune 89            "       "  Neptune 149 

Venus  by  Mercury 93        Mars  by  Mercury 152 

"   Earth .100            "       "  Venus : 156 

"        "  Mars 104            "       "  Earth 160 

"        "  Jupiter 108    .        "       "  Jupiter 164 

"  Saturn 112            "       "  Saturn 168 

"  Uranus 116            "       "  Uranus 173 

"  Neptune 120            "       "  Neptune 176 

11.  The  final  values  of  the  perturbations 179 

12.  Comparison  with  the  results  of  observation 

13.  Comparison  with  SEELIGER'S  hypothesis  on  the  constitution  of  the  Zodiacal  Light 185 


.    37 


1.  INTRODUCTION. 

The  usual  method  of  determining  the  secular  variations  of  the  elements  of  any 
planet  is  the  well-known  one  based  upon  the  development  of  the  perturbing  function 
into  an  infinite  series  whose  successive  terms  involve  continually  higher  powers  of 
the  eccentricities  and  the  mutual  inclination.  This  method  possesses  two  advantages. 
The  first  is  that  when  an  extreme  degree  of  accuracy  is  not  required,  so  that  higher 
terms  of  the  development  may  be  disregarded,  it  is  the  simplest  method  available; 
and,  in  the  second  place,  since  the  coefficients  of  all  terms  are  general  literal  expres- 
sions, the  change  produced  in  the  value  of  any  variation  by  a  change  in  the  assumed 
values  of  one  or  more  of  the  elements  can  readily  be  ascertained  by  a  simple  substi- 
tution of  the  more  accurate  values.  On  the  other  hand,  this  method  possesses  the 
disadvantage  that  the  complexity  of  the  expansion  grows  rapidly  greater  as  the  order 
of  the  included  terms  is  increased,  so  that  a  slight  increase  in  the  desired  accuracy 
greatly  increases  the  labor  of  the  computation. 

The  integral  methods,  founded  upon  the  celebrated  theorem  of  GAUSS(I),*  are 
wholly  free  from  this  latter  disadvantage,  for  if  it  is  desired  to  include  all  terms  to 
the  twenty  fourth  order  this  can  be  done  by  a  computation  which  is  less  than  twice 
as  long  as  that  required  when  the  approximation  is  stopped  at  terms  of  the  eleventh 
order.  But  the  integral  method,  though  thus  extremely  accurate,  leads  only  to  the 
numerical  values  of  the  variations  dependent  upon  the  values  of  the  elements  assumed ; 
if  they  are  desired  for  some  other  epoch  at  which  the  various  elements  possess  different 
values  from  those  adopted,  or  if  an  improved  value  of  any  of  the  elements  becomes 
known,  they  can  only  be  found  by  an  entire  repetition  of  the  computation. 

The  only  determinations  of  the  secular  perturbations  of  the  four  inner  planets 
which  are  in  any  sense  modern  ones  are  the  classic  investigation  of  LE  VERRIER(T) 
and  the  computation  of  NEWCOMB(15>.  The  latter  furnishes  the  most  accurate  values 
of  these  variations  so  far  determined ;  the  series  were  extended  to  terms  of  the  eighth 
order,  only  those  terms  of  this  order  being  included,  however,  which  seemed  likely 
to  be  most  important,  and  in  some  cases  terms  of  the  tenth  order  were  included, 
though  usually  by  induction  merely. 

In  both  of  the  above  computations  the  usual  expansion  into  an  infinite  series 
was  employed.  As  the  GAUSSIAN  method  is  so  extremely  accurate,  and  as  its  formulas 
throughout  are  wholly  different  from  those  hitherto  employed,  it  seemed  that  an 

*  These  symbols  wherever  they  occur  refer  to  the  list  of  titles  at  the  end  of  the  present  paper. 

39 


40  THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 

application  of  it  to  a  re-determination  of  these  variations  based  upon  the  most  ac- 
curate values  of  the  several  elements  now  obtainable  would  be  of  value.  The  results 
of  this  work  will  be  found  in  the  following  pages;  the  final  comparison  with  the 
earlier  results  is  given  in  Article  11,  and  the  comparison  with  the  results  of  observa- 
tion in  Articles  12  and  13.  The  epoch  throughout  is  1850.0,  G.  M.  T. 

In  the  four  following  articles  an  attempt  is  made  to  state  briefly  the  essential 
features  of  the  various  methods  of  computing  secular  variations  which  are  founded 
on  GAUSS'S  theorem,  but  for  a  detailed  account  of  the  long  and  often  complex  trans- 
formations which  are  involved,  the  original  papers  must  be  consulted. 

2.  THE  METHOD  OF  GAUSS. 

The  equations  which  express  the  complete  variations  of  the  elements  of  the 
orbit  of  any  body  revolving  about  the  sun  when  it  is  disturbed  in  its  motion  by  the 
presence  of  a  third  body,  may,  as  is  well  known,  be  put  in  a  variety  of  different 
forms;  the  form  selected  as  the  basis  for  all  developments  founded  on  GAUSS'S  method0' 
is  that  in  which  three  rectangular  components  of  the  disturbing  force  enter  into 
the  expressions  for  the  differential  coefficients.  Thus,  if  R  denote  the  component 
lying  in  the  direction  of  the  radius  vector  of  the  disturbed  body,  positive  outward 
from  the  sun;  S,  the  component  lying  in  the  plane  of  the  orbit  of  the  disturbed  body 
and  perpendicular  to  the  radius  vector,  positive  in  the  direction  of  motion;  and  W, 
the  component  perpendicular  to  this  plane  and  positive  northward,  we  will  have 
for  the  variation  of  the  eccentricity  of  the  orbit  of  the  disturbed  body, 

de       tfn  cos<p 

dt  =  fe'(l  +  m)  ^sm         +  (-cos  "  +  cos     ^   J' 

with  similar  expressions  for  the  variations  of  the  six  remaining  elements.* 

In  the  original  memoir  of  GAUSS  the  determination  of  the  secular  terms  of 
these  expressions  was  given  a  geometrical  aspect.  Thus,  since  each  variation  may 
obviously  be  expressed  in  terms  of  the  two  single  variables  M  and  M' ,  the  secular 
term  in  question  will  be  that  given  by  the  equation, 


[de-\          1     r2'  ("'de  ,..,.., 
[dt  1=4^1    1     dtdMdM' 


*  The  usual  notation  is  adopted  throughout.  Thus  a,  e  =  sin  ip,  i,  12,  ir,  n,  and  L  are  respectively  the  half  major 
axis,  the  eccentricity,  the  inclination,  the  longitude  of  the  ascending  node,  the  longitude  of  perihelion,  the  mean  motion 
and  the  longitude  at  the  epoch  of  the  disturbed  body;  M,  E,  v  and  r  are  respectively  its  mean,  eccentric  and  true  anomalies 
and  its  radius  vector,  m0  is  its  mass,  k*  is  the  mass  of  the  sun,  and  m0  =  >nk2.  The  same  letters  with  accents  refer  to 
the  disturbing  body. 

Watson,  Theoretical  Astronomy,  pp.  516-523;  Oppolzer,  Lehrbuch  zur  Bahnbestimmung,  Vol.  II,  p.  213;  Tisserand, 
Mecanique  Celeste,  Vol.  I,  pp.  431-433,  etc.  The  final  forms  of  the  equations  expressing  the  other  variations  may  be 
inferred  from  those  stated  at  the  end  of  Article  7. 


OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  41 

and  this  is  the  same  as, 

[de~\  a2ncos<f      f2"  f  1    f  2ir  _  „     1    f2'  „  ,  ,,  ,  1  ,  ,, 

=  o  70/1  sm  t>-  5-         /MM'  +  (cos  v  +  cosE)-  ^-        SdM'    dM, 

\_dt  !joo       27r£2(l  +  m)  J0      L  2irJ0  2irJ0  J 


since  the  variable  of  the  first  integration  enters  the  expression  only  through  R  and  S. 
In  the  equation  as  thus  written  R  and  S  are  supposed  to  contain  the  mass,  ra0',  as  a 
factor  so  that  if  Ri  and  Si  are  the  corresponding  values  produced  by  a  unit  mass, 
R  =  m0'Ri  and  S  =  m0'Si. 

If  we  now  imagine  an  infinitely  thin  elliptic  ring  which  coincides  with  the  orbit 
of  m',  whose  total  mass  is  equal  to  the  mass  ra</,  and  the  density  of  any  portion  of 
which  is  proportional  to  the  time  occupied  by  m'  in  describing  that  portion  of  its 
orbit,  we  will  have  for  the  three  components  of  the  attraction  exerted  by  any  portion 
dm0f, 

R.dmo',        Sidm'o,        and 


and  integrating  about  the  entire  ring,  we  find  for  the  complete  components, 

flit  /*2n-  /»2jr 

Rdmo,          I     Sidmo',        and          I      W4m<>'- 
Jo  Jo 

But  by  the  conditions, 


dt       dM' 


OTo'  "  T          2;r  ' 

and  hence  the  components  are, 

J         rtn  1         r>2«  1        /*2ir 

m^'RidM',         5-         mQ'StdM'        and         ~-         mo'T^id/lf, 

<i7T  J0  ^TT  J0  /7T  J0 

which  are  identical  with 

^  f '"  fldM',         ^  (^  SdM'        and         g.  if"  W^dM'. 

Thus  the  expressions  giving  the  secular  variations  are  seen  to  be  the  same  whether 
these  are  derived  from  the  moving  planet  or  from  the  elliptic  ring.* 

The  work  of  GAUSS  contains  no  application  to  the  determination  of  secular  vari- 
ations nor  are  all  the  formulas  necessary  for  this  purpose  there  developed;  the  first 
integration  alone  is  effected,  and  it  is  shown  that  by  changing  first  to  the  variable 
E'  and  afterward  introducing  a  new  variable,  T,  each  of  the  complicated  integrals 
may  be  made  to  depend  upon  elliptic  integrals  whose  values  GAUSS  obtained  by  the 
introduction  of  a  new  algorithm  called  by  him  the  Arithmetico-geometrical  mean. 

*  Other  interesting  geometrical  aspects  of  the  problem  are  treated  by  Bour  <5),  Hill  <s"'  (38),  and  Halphen  <28),  but 
for  brevity  a  detailed  account  of  these  is  here  omitted. 


42  THE    SECULAR   VARIATIONS    OF   THE    ELEMENTS 

The  first  application  of  GAUSS'S  method  was  made  by  NicoLAi(2),  who  determined 
by  it  the  secular  variations  of  the  Earth's  orbit,  but  the  results  only  were  published.* 
The  first  development  of  the  method  is  by  CLAUSEN  (3)  who  also  applied  it  to  a  determi- 
nation of  the  perturbations  of  Tuttle's  Comet  produced  by  the  action  of  Jupiter (4), 
dividing  the  disturbed  orbit  into  120  parts  with  reference  to  the  true  anomaly.  It 
was  next,  in  1867,  applied  by  ADAMS((!)  to  the  orbit  of  the  November  meteors  with  a 
special  view  to  ascertaining  the  cause  of  the  steady  progression  of  the  node  of  the 
orbit,  but  in  this  investigation  certain  small  terms  were  neglected  by  ADAMS  and  the 
solution  of  a  fundamental  cubic  equation  which  occurs  in  the  original  method  was  in 
this  manner  avoided. 

No  further  applications  of  GAUSS'S  method  seem  to  have  been  made  until  after 
the  publication  of  HILL'S  extensive  development (8)  and  modifications  of  it  in  1882. 

3.    HILL'S  FIRST  MODIFICATION  OF  GAUSS'S  METHOD. 

Although  the  first  of  the  above  integrations  may  be  rigorously  effected,  the 
value  of  the  second  must  be  approximated  to  by  a  mechanical  quadrature  about  the 
orbit  of  m,  a  greater  or  less  number  of  terms  being  employed  in  the  quadrature 
according  as  the  disturbed  orbit  is  more  or  less  eccentric.  Since  either  the  true, 
eccentric,  or  mean  anomalies  may  be  selected  as  the  variables,  it  becomes  of  im- 
portance to  decide  which  of  these  must  be  chosen  in  order  to  render  the  quadrature 
most  accurate.  It  is  readily  proved!  that  the  inequalities  of  distribution  of  a  series 
of  points  on  an  elliptic  orbit  corresponding  to  a  series  of  equidistant  values  of  the 
eccentric  anomaly  are  of  the  order  of  the  square  of  the  eccentricity  while  for  the 
other  two  anomalies  they  are  of  the  order  of  the  first  power  of  this  quantity,  and 
therefore  HILL  has  employed  the  eccentric  anomalies  throughout  his  development, 
although  SEELiGER(9)  showed  that  a  still  higher  accuracy  will  be  obtained  if  the  true 
anomalies  are  chosen. 

If,  therefore,  we  decide  to  make  the  integrations  with  reference  to  the  eccentric 
anomalies,  we  will  obtain,  since 

dM  =  ^dE,        AM'  =--  r-,dE',        and        r'  =  a' (I  -  e'  cos  E'), 

[de~\          ncos<p        I     f2"  f 2ir . 
j,       =  ,,,,       — r  •  -r—»  \  sin  v  •  Kar(I  —  e  cos  E) 

L^JOO          &2(1    +  TO)      47T2J0        J0 

+  (cos  v  +  cos  E)  •  Sar(l  -  e'  cos  E')]dEdE'dt, 

*See  Article  11. 

t  See  Tisserand's  Mecanique  Celeste,  Vol.  I,  page  442, 


OF   THE    ORBITS   OF   THE  FOUR   INNER    PLANETS.  43 

and  writing, 

1      r2"  nr  1      C2"  nr 

R0  =  ^     I      —,R(l-e'  cosE')dE',  S0  =  ^  —,S(l-e'  cos  E')dE', 

JIT  JQ     TOO  zir  JQ     mo 


the  expression  for  the  secular  variation  will  become, 

[del          m'n  1     T2" 

'-  cos  <p  •  JT-          [sin  v-Ro  +  (cos  v  +  cos  E)Su]dE. 
at  Joo       1  -f-  TO  ZTT  J0 

In  order  to  find  the  values  of  R0,  S0,  and  W0,  it  is  first  necessary  to  express 
R,  S  and  W  in  terms  of  E'  '.  For  this  purpose  that  part  of  the  disturbing  force  arising 
from  the  action  of  the  disturbing  planet  upon  the  sun  need  not  be  included,  for  it  is 
known  that  this  has  no  secular  term.|  Considering  therefore  only  the  action  of  m' 
upon  m,  it  is  evident  from  a  figure  that  R,  S  and  W  will  have  the  values, 


_  mo'  I  r>  cos  #  —  r  1 

I  / 

_  TOO'     r'  sin  &  cos  y 


TT7  °         ' 

:  -^r sm  y> 

and  also  that 

A-'  =  r2  -  2rr'  cos  «  +  r'2, 

in  which  &  is  the  angle  included  between  the  radii  vectores,  A  is  the  distance  between 
the  two  bodies,  and  y  is  the  inclination  of  the  plane  which  includes  r  and  r'  to  the 
plane  of  the  orbit  of  the  disturbed  body. 

If  n  and  n'  denote  the  angular  distances  respectively  of  the  perihelia  of  the  two 
orbits  from  the  ascending  node  of  the  orbit  of  m'  upon  the  orbit  of  m,  and  if  /  be  their 
mutual  inclination,  we  will  have, 

cos  0  =  cos  (v  +  n)  cos  (vr  +  II')  +  sin  (v  +  U)  sin  (v'  +  n')  cos  /, 
sin  0  cos  7  =  —  sin  (v  +  n)  cos  (v'  +  II')  +  cos  (v  +  II)  sin  (v'  +  II')  cos  I, 

sin     sin  7  =  sin  /  sin  (v'  +  n') 

The  values  of  n,  n',  and  /  are  obtained  from  the  original  elements  by  a  direct 
solution  of  the  spherical  triangle  whose  sides  are  n  and  n',  and  in  which  the  angle 
included  between  these  sides  is  I.  (See  Article  7.) 

tSee  Hill's  "On  Gauss's  Method  <8>,  ,  .  ,"  page  321. 


44  THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 

If  we  now  eliminate  v'  from  the  above  expressions  by  the  equations, 

r'  cos  v'  =  a'  (cos  E'   -  e'),        r'  sin  v'  =  a'  cos  <p'  sin  E',        r'  =  a'  (1  —  e'  cos  E'), 

the  resulting  equations  giving  R,  S,  W,  and  A  will  be  expressed  wholly  in  terms  of 
the  variable  E'  .  In  order  to  simplify  these  results,  we  assume  certain  new  auxiliaries 
defined  by  the  equations, 

k  cos  (.K-n)  =  cos  IT,       k  sin  (K-U)  =-  cos  /  sin  IT,       k'  cos  (K'-Il)  =  cos  7  cos  II', 

k'  sin  (K'  -  II)  =     -  sin  II', 
A  =  rz  +  2ka'e'r  cos  (v  +  K)  +  a'\ 
B  cos  e  =  ka'r  cos  (v  +  K)  +  o'V, 
B  sin  e  =  /r'a'  cos  <p'  •  r  sin  (v  +  K'), 

Ac  =  ka'  cos  (v  +  K),  As  =  k'a'  cos  <?'  sin  (v  +  K') 

Bc  =  -  ka'  sin  (v  +  K),  Bs  =  k'a'  cos  v'  cos  (v  +  A'') 

Cc  =  a'  sin  n'  sin  /,  Cs  =  a'  cos  <p'  cos  II'  sin  /. 

C  =  a'V3, 
when  the  desired  expressions  become, 

-*•,  R  =  4,  (cos  E'  -  e')  +  As  sin  E'  -  r 

W?o 

•^U  =  5c(cos  £'  -  e')  +  58  sin  £' 

W?0 

~  W  =  Cc(cos  £'  -  e')  +  C,  sin  #' 


2 


A2  =  A-2B  cos  (£'  -  e)  +  C  cos 

In  order  to  effect  the  integrations,  GAUSS  here  introduced  a  new  variable,  T, 
connected  •  with  E'  by  the  relations, 

N  sin  E'  =  a  +  a'  sin  T  +  a"  cos  T 
N  cos  E'  =  /3  +  0'  sin  71  +  0"  cos  7T 
N  =  7  +  7'  sin  T  +  j"  cos  77, 

the  quantities  «,  a',  a",  0,  0'  .  .  .  being  subject  to  the  conditions  that 
(N  sin  #')2  +  (N  cos  #')2  -  JV2        and        sin2  T  +  cos2  T  -  1 

shall  be  identically  zero,  and  also  being  so  chosen  that  the  coefficients  of  sin  T,  cos  T, 
and  sin  T  cos  T  shall  vanish  in  the  expression  2V2  A2  which  therefore  must  take  the 

form, 

G  -  G'  sin2  T  +  G"  cos2  T. 

From  these  conditions  it  is  derived  that  the  coefficients  G,  G'  and  —  G"  in  the  trans- 


OF   THE    ORBITS    OF   THE    FOUR   INNER   PLANETS.  45 

formed  expression  for  JV2A2  must  severally  satisfy  the  cubic  equation, 

x(x  -  A)(x  +  C)  +  B2x  +  B2C  sin2  e  =  0, 

and  hence  that  they  must  be  the  roots  of  this  equation.  By  substituting  for  x  the 
successive  values,  --  (7,  0,  a'2  cos2  <p'  and  +  A,  the  first  member  is  seen  to  take  in 
succession  the  corresponding  values, 

-  B2C  cos2  e 
+  B2C  sin2  e 

-  a'4  cos2  <p'  •  r2  sin2  1  sin2  (v  +  II) 
+  B*-(A  +  C  sin2  e). 

Since,  even  when  cos  (v  +  K)  has  its  maximum  negative  value,  the  value  of  A 
exceeds  that  of  (r  —  a')2,  it  is  evident  that  A  is  always  positive,  and  therefore  that 
the  above  equation  has  one  negative  root  which  lies  between  —  C  and  0,  one  positive 
root  lying  between  0  and  a'2  cos2  <p',  and  that  the  third  root  lies  between  this  value 
and  +  A.  The  roots  are  represented  by  —  G",  G',  and  G,  respectively,  and  thus 
G",  G'  and  G  are  always  positive  quantities,  the  last  being  the  largest  and  the  first 
the  smallest  except  when  <p'  exceeds  45°,  a  case  not  met  with  in  any  of  the  planetary 
orbits. 

Since  a,  j3,  y,  a',  0'  .  .  .  must  retain  the  same  values  whatever  the  values  of 
E'  and  T,  we  may,  by  writing  the  equations  arising  from  the  three  conditions  above 
stated  and  equating  the  coefficients  of  the  like  terms  in  the  two  members,  obtain 
a  series  of  equations  which  are  sufficient  for  the  determination  of  these  quantities 
in  terms  of  G,  G',  G"  and  the  other  known  auxiliaries.  Upon  substituting  the  resulting 
expressions  for  sin  E'  and  cos  E'  in  the  equations  defining  R0,  S0,  and  T^o,  and  noticing 
that  JV2A2  may  be  written, 

G  -  G'  sin2  T  +  G"  cos2  T  =  (G'  +  G"}  1  1  -  ^-^'sin2  T  }, 

I  tr  +  tr  J 

we  obtain  each  of  the  components  in  the  form, 

ms  sin2  T  +  mc  cos2  T 
or 


1    I'"" 
e  •=  «- 

If  we  now  write, 


G 


/•I   I   fin  —        -          7, 
and  consider  that  from  LANDEN'S  well-known  transformation, 


46  THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


c~ 


and  also  notice  that 

r'2  dr 

Jo     (l-c2sin* 

*       sin2  TdT 


cos*  TdT  „ 


![„/     TT\ 

=  ?L  vc>2/ 


it  is  evident  that  each  of  the  above  three  integrals  becomes  expressible  wholly  in 
terms  of  the  rapidly  convergent  series  of  LANDEN. 

For  the  purposes  of  the  present  computation  HILL(S)  has  computed  to  ten  places 
the  logarithms  of  the  quantities 


K0  =  sec2  0  •  KL,        L'0  =  L  ~         B,        and        Nu  =  sec2  0  •  (1  +  £„'), 


and  these  correct  to  eight  places  are  tabulated  at  intervals  of  one  tenth  of  a  degree 
for  all  values  of  0  from  0  =  0°  to  0  =  50°. 

From  a  direct  substitution  it  is  now  seen  that  the  final  resulting  values  of  Ro, 
S0  and  W0  are  as  follows,  in  which  the  symbols  N,  P,  Q,  etc.,  are  written  for  abbrevi- 
ation and  have  the  meanings  stated  in  Article  7  : 

7?0  =  -  N  -  QG'  +  VJS, 
S0  =  PF*  +  VJ, 
W0  =  PF,  +  VJ3 

The  integration  with  respect  to  E'  having  been  thus  entirely  completed,  that 
in  regard  to  E  is  effected  by  mechanical  quadratures.  Since  each  variation  is  a 
function  of  E  alone,  it  follows  by  the  principles  of  quadratures  that  if  any  one  of  them 
be  expanded  into  a  periodic  series  involving  the  sines  and  cosines  of  E  and  its  multiples, 

the  secular  term  of  the  series,  which  is  rigorously  equal  to  |ir  I    f(E)dE,  may  be 

I/O 

also  obtained  by  forming  the  values  of  f(E)  for  2j  equidistant  values  of  E,  from 
E  =  0°  to  E  =  360°,  and  dividing  the  sum  by  2j.     The  expression  thus  obtained, 


will  be  subject  only  to  the  error  involved  in  dropping  those  terms  which  contain  a 
multiple  of  E  not  lower  than  2j.  An  inspection  of  the  known  forms  of  the  series 
which  express  the  variations  renders  it  evident  that  the  error  thus  committed  is  of 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS.  47 

the  order  2j  in  terms  of  the  eccentricities  and  mutual  inclinations  of  the  orbits  except 
in  the  one  case  of  the  variation  of  the  Mean  Longitude,  in  which,  as  this  variation 
depends  wholly  upon  the  expansion  of  —  2(r/a)JRo>  it  is  of  the  order  2j  +  1. 

The  resulting  equations  giving  the  values  of  all  the  secular  variations  are  those 
stated  in  Article  7. 

4.    HILL'S  SECOND  MODIFICATION  OF  GAUSS'S  METHOD.    THE  WORK 

OF  CALLANDREAU  AND  INNES. 

In  HILL'S  second  modification  of  GAUSS'S  method(8>,  the  well-known  expressions 
for  the  roots  of  a  cubic  equation  when  this  is  solved  by  the  trigonometric  method  are 
introduced,  and  thus,  throughout  the  integrals,  the  quantities  p,  q  and  0'  occur  instead 
of  the  roots  G,  G'  and  G",  the  equations  connecting  these  quantities  being, 


G  =  29sin60°  -         +  P,         G'  =  2gsin 
G"  =  2g  sin  (60°+  I')  -p. 

It  was  shown  in  GAUSS'S  original  memoir(1)  that 

dT 


f 


(m2  cos2  T  +  n2  sin2  T)  *      J0      (m'2  COS2 


if  m'  =  \(m  +  w)  and  w'  =  V  mn,  and  that  by  repeating  this  transformation  by  the 
employment  of  the  equations, 

m"  =  i(m>  +  n'),  n"  =  Jrnfri, 

m'"  =  \(m"  +  n"),        n'"  =  JriW, 
etc.  etc., 

m(*>  and  n(k)  very  rapidly  approach  a  single  limit,  p.,  which  GAUSS  named  the  Arith- 
metico-geometrical  Mean.  It  thus  follows  that  our  first  integral  is  equal  to  7r/2^, 
and  that  integrals  of  the  form 

p    (sin2  T  -  cos2  T)dT 
J0    (m2  cos2  T  +  n2  sin2  T)* 

become  equal  to  ir/2  •  w/ju  in  which  w  is  a  very  rapidly  converging  series  involving 
m,  n,  m',  n',  etc.,  in  its  successive  terms. 

The  integral  expressions  which  actually  enter  into  the  equations  for  .R0,  S0,  and 
W  o  are 


(f\f  \  -/o 

60°  -  Q  )  -    r  -r^ 
3  /        4     (m2  — 


n2) 


48  THE   SECULAR   VARIATIONS   OF   THE    ELEMENTS 

V3       w  sin  6' 


in  which 

tf 

A 
o 

and 


the  values  of  #0,  £0,  and  T^0  being  connected  by  comparatively  simple  relations  with 
these  quantities  and  with  known  auxiliaries. 

HILL  accordingly  suggested  that  tables  of  these  functions  should  be  computed, 
and  this  was  first  done  by  MONS.  0.  CALLANDREAU(I:!>  who  however  adopted  as  an 
argument  the  quantity  a  defined  by  the  relation 


1 
1  — 


cos 


1  +  a    •  6' 

V  COSQ 

*  <5 

and  tabulated  the  logarithms  of  the  functions  r«4n4x(0')  and  <K0')  •*-  x(0')  at  intervals 
of  0.001  from  «  =  0.000  to  a  =  0.400;  of  0.002  from  «  =  0.400  to  a  =  0.600  and  of 
0.005  from  this  point  to  the  extreme  value,  «  =  1.000.  This  paper  repeats  the 
derivation  of  all  formulas  necessary  when  the  second  method  alone  is  employed, 
essentially  as  this  was  given  by  HILL,  and  also  contains  a  direct  proof  that  R0,  S0, 
and  W0  can  be  expressed  wholly  in  terms  of  the  complete  elliptic  integrals,  F  and  E. 

Similar  tables  were  also  computed  by  MR.  R.  T.  A.  lNNES(22),  the  functions 
here  tabulated  being  (1  --  «)/(!  +  «4)  •  \l/(6')  and  \l/(6')  +  *(0')  to  the  argument  0' 
at  intervals  of  one  degree,  from  0'  =  —  90°  to  0'  =  +  90°. 

Whether  the  first  or  second  methods  be  employed,  the  values  of  the  integrals 
involved  may  also,  as  was  pointed  out  by  HiLL(38),  be  approximated  to  with  great 
rapidity  by  the  use  of  JACOBI'S  Nome,  q  (American  Journal  of  Mathematics,  Vol.  23, 
page  321.  In  the  Astronomical  Journal,  No.  511,  a  brief  application  is  given  to  a  case 
in  the  action  of  Venus  on  the  Earth).  This  function  is  defined  by  the  equation, 

— 
q  =  e~'x, 

in  which  K'  is  the  complete  elliptic  integral  of  the  first  kind  complementary  to  K, 
from  which  there  may  be  derived, 

KE=(1+  3V  +  5V  •  •  •)  -  U  +  <?2  +  <?6  • '  •)• 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS.  49 

The  values  of  log  [(//tan2  £0]  computed  to  ten  decimal  places  for  each  degree  of  8 
from  6  =  0°  to  6  =  45°  are  given  by  lNNES(39).  When  0  exceeds  45°,  the  values  of  K 
and  E  are  readily  obtained  from  their  expressions  in  terms  of  the  complementary 
complete  integrals  whose  moduli  are  sin  (ir/2  —  0),  and  to  which  the  table  is  therefore 
directly  applicable. 

Lastly,  in  the  second  method,  HILL  recommends  that  the  quadratures  be  per- 
formed upon  the  quantities  a/r  •  R0,  a/r  •  S0  and  a/r  •  W0  directly,  all  constant  and 
evanescent  factors  which  appear  in  the  expressions  for  the  variations  being  removed 
from  under  the  integral  signs  and  reserved  until  the  integration  has  been  completed. 

5.    THE  METHOD  OF  HALPHEN  AND  ITS  MODIFICATIONS  BY  ARNDT 

AND  INNES. 

It  was  first  pointed  out  by  BRUNS(29)  that  the  periods  of  the  elliptic  functions 
of  the  first  and  second  integrals  can  be  evaluated  without  a  knowledge  of  the  three 
roots,  but  it  was  HALPHEN (28)  who  first  applied  this  remarkably  elegant  method  of 
analysis  to  the  present  problem.  It  was  shown  by  him  that  if  o>  and  17  are  the  two 
periods  in  question,  then  R0,  So,  and  W0  may  be  obtained  in  the  form  aw  +  br\,  in 
which  a  and  b  are  rational  functions  of  the  coefficients  of  the  cubic  equation  and  w 
and  TJ  are  expressible  in  terms  of  certain  hyper-geometric  series  in  which  the  common 
variable  is  an  absolute  invariant  of  the  elliptic  functions. 

The  three  integrals  entering  into  the  problem  have  the  form, 


/" 

Jo 


IdT 


71)3' 


in  which  /  has  the  values  1,  sin2  T  and  cos2  T,  respectively,  in  the  three  cases;  by 
introducing  the  new  variable,  s,  defined  by  the  relation, 

G  +  G"    s-G' 
G'  +  G"  '  s  -  G  ' 
these  become, 

G>  +  G"   r-o"  te  G  +  G"   r-o»  ds 

~Jl    ~^s(s  'G}>      ~2~^~i    v!(s' 

and 


n 
respectively,  in  which 

f/~<    i    r<u\(r<       r«\ir<t   i    r»>\ 
n  =  ((j  -\-  (j  ;  ((JT  —  Cr )  (u    -f-  Cr  ; 

and 

S  =  -  4(«  -  G)(s  -  G')(s  +  (?")• 


50  THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 

Introducing  the  WEIERSTRASSIAN  r  function  through  the  relation 

C  ds 

U  =  % 

u  being  the  elliptic  integral  of  the  first  kind  and  61,  e2,  and  e3  the  roots  of  the  cubic 
equation  increased  by  one  third  of  the  coefficient  of  x2,  and  considering  that  from  the 
theory  of  these  functions, 

s  -  G  =  r(u)  -  ei,         T(w)  =  d,         r(w  +  w')  =  e2,     and         T(w')  =  e3, 
the  first  integral  will  become, 

2  —  ^~  f"  "  fr(«)  -  e'ld«  =  2\e^  +  ~  («  +  «')  -  -  '  w'l 
Ju  L  °"  f        J 

ff  and  o-'  being  the  second  WEIERSTRASSIAN  functions,  which  are  connected  with 
the  periods,  w  and  rj,  by  the  equations, 

—  (<•>  +  «')  =  77  +  ?;';        -co'  =  •)?'. 

The  three  integrals  consequently  take  the  final  forms, 

_G'  +  G"  nG  +  G",  nG'-G, 

2  ~  -  (eico  +  77) ;   2  -          —  (^co  +  ?;),    and    2  -         ~-  (630)  +  77). 

fv  ra  71 

A  direct  substitution  of  these  expressions  for  the  integrals  in  the  equations  which 
define  .R0>  So  and  Wo  leads,  after  some  reduction,  to  forms  which  are  seen  to  contain 
only  these  integrals  themselves,  the  coefficients  of  the  cubic  equation  with  other 
known  auxiliaries,  and  the  quantity  n.  But  if,  for  brevity,  we  write  the  original 
cubic  equation  in  the  form, 

x3  -  P,x2  +  P*x  -  P,  =  0, 
and  let 

X  =  P,1  -  3P2        and        p  =  PjP2  -  9P3, 

then  the  invariants,  g2  and  g3,  and  the  absolute  invariant,  g,  will  have  the  values, 

to  =  |X;         0i  =  A(2PiX  -  3p),         and         g  =  023  •*•  27032, 

and  w  will  be  given  by, 

n~  =  leC^3  -  27^32), 

in  which  the  last  factor  is  the  discriminant.  Thus,  except  for  <o  and  TJ,  our  final 
expressions  are  obtained  wholly  in  terms  of  the  coefficients  of  the  cubic  equation, 
and  a  knowledge  of  the  roots  becomes  unnecessary. 

In  the  paper  by  BRUNS,  before  referred  to,  it  is  shown  that  w  and  i\  are  directly 


OF   THE   ORBITS   OF   THE    FOUR   INNER   PLANETS.  51 

expressible  in  terms  of  a  hyper-geometric  series  whose  variable  is  the  absolute  invari- 
ant, g.  By  a  simple  transformation  the  relations  may  be  placed  in  the  following 
forms,  which  are  more  convenient  in  practical  application. 

A.  ,  o- 

'  12'  l>       g 


DR.  Louis  ARNDTi30)  has  fully  developed  this  method,  deriving  all  the  formu- 
las necessary  for  its  application  and  stating  tables  for  F(w)  and  F(T))  for  values  of 
(g  -  l)/0  from  (g  -  l)/g  =  0.000  to  (g  -  !),/</  =  0.980,  the  interval  being  0.001. 

In  a  recent  paper  by  INNES(SI)  the  complete  formulas  for  this  method  are  derived 
when  the  quadrature  is  applied  directly  to  the  expressions  (a/r)R0)  (a/r)SQ  and 
(a}r)Wo,  as  suggested  in  the  second  method  of  HILL.  The  development  is  nearly 
identical  with  that  of  ARNDT  except  that  the  forms  of  the  hyper-geometric  series 
are  slightly  changed,  the  variable, 

,  *      Vg  -  1 
sin2  2  =  — *—?=- 

being  preferred.     The  values  of  the  logarithms  of 

F  («)  =  F(!,  I  2,  sin2  -}        and        F  (77)  =  F  (-  g,  |  2,  sin2  *  Y 

were  published  by  MR.  FRANK  RoBBiNS(32)  for  all  values  of  i,  at  intervals  of  one 
degree  from  i  =  1°  to  i  =  90°,  the  computation  having  been  made  to  ten  places  and 
published  to  seven,  and  these  tables,  computed  with  seven  place  logarithms,  have 
been  extended  from  i  =  90°  to  i  =  180°  by  MR.  C.  J.  MERFiELD(33). 

Although  the  preceding  methods  are  of  great  mathematical  elegance,  it  is  doubt- 
ful whether  their  formulas  lead  to  so  accurate  results  as  those  of  HILL'S  first  method 
when  seven  place  logarithms  are  employed.  (See  the  computations  of  Jupiter  on 
Mars(24)  and  of  Saturn  on  Mars(25>,  Article  10.)  Moreover,  when  the  method  is 
applied  which  is  explained  in  the  computation  of  Jupiter  on  Mercury  (Article  10), 
the  roots  of  the  cubic  equation  are  so  readily  obtained  that  the  avoidance  of  its 
solution  becomes  a  matter  of  no  practical  importance.  Accordingly  HILL'S  first 
modification  of  GAUSS'S  method  has  been  employed  throughout  all  of  the  following 
computation. 


THE  COMPUTATION. 


6.     THE  ELEMENTS  OF  THE  ORBITS  AND  THE  ADOPTED  MASSES. 

The  values  adopted  for  the  elements  of  the  several  orbits,  to  serve  as  the  basis 
for  this  computation,  were  taken  in  each  case  from  HILL'S  "New  Theory  of  Jupiter 
and  Saturn." (16)  Those  of  the  four  inner  planets  will  be  found  on  page  192;  those 
of  Jupiter  and  Saturn  on  page  558;  of  Uranus  on  page  109,  and  of  Neptune  on  page 
161.  The  epoch  throughout  is  1850.0  G.  M.  T. 

The  values  of  the  masses  finally  selected  by  HILL,  and  here  adopted,  will  be 
found  on  page  554  for  Mercury,  Venus  and  the  Earth;  on  page  192  for  Mars;  on  page 
19  for  Jupiter  and  Saturn,  and  on  page  161  for  Neptune.  The  mass  of  Uranus  as 
stated  in  the  "New  Theory  "  is  1  -r-  22640,  but  at  DR.  HILL'S  suggestion  this  is  here 
diminished  to  1  -4-  22800,  (A.  J.,  No.  316).  The  value  assumed  for  the  mass  of 
Mercury  when  the  first  of  these  computations  were  made  was  1  -r-  5000000,  but  all 
of 'the  results  are  here  changed  to  agree  with  the  value  1  -r-  7500000  stated  below.  It 
seems  not  improbable  that  even  this  latter  fraction  is  too  large,  but  the  true  value  of 
this  element  is  still  very  uncertain. 


X 

i 

S2        I       e 

n 

Mercury 
Venus 
Earth 
Mars 
Jupiter 
Saturn 
Uranus 
Neptune 

o    /    // 

75   7  13.62 
129  27  42.83 
100  21  39.73 
333  17  51.74 
11  54  31.67 
90   6  41.37 
168  15   6.70 
43  17  30.30 

o    /    // 

7   0   7.71 
3  23  35.01 
0   0   0.00 
1  51   2.24 
1  18  42.10 
2  29  40.19 
0  46  20.54 
1  47   1.68 

46°  33'  8^63 
75  19  53.08 

48  23  54.59 
98  56  19.79 
112  20  49.05 
73  14   8.00 
130   7  31.83 

0.20560476 
0.00684311 
0.01677114 
0.09326803 
0.04825511 
0.05606025 
0.0469236 
0.0084962 

5381016^260 
2106641.357 
1295977.416 
689050.784 
109256.626 
43996.21506 
.15425.752 
7864.935 

logo 

1-j-m 

Mercury 

9.5878217 

7  500  000 

Venus 

9.8593378 

408  134 

Earth 

0.0000000 

327  000 

Mars 

0.1828971 

3  093  500 

Jupiter 

0.7162374 

1  047.879 

Saturn 

0.9794956 

3  501.6 

Uranus 

1.2831044 

22  800 

Neptune 

1.4781414 

19  700 

52 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS.  53 

7.    THE  FORMULAS  EMPLOYED  IN  THE  COMPUTATION. 

The  following  formulas  are  written  in  the  order  in  which  they  were  applied. 
When  the  right  hand  member  appears  in  two  different  forms,  one  of  these  was  used 
in  the  first  computation  and  the  other  in  the  duplication,  though  sometimes  other 
obvious  modifications  were  made  use  of  in  the  several  cases  differing  from  those  which 
are  here  written. 

The  values  of  /,  n,  and  n'  were  obtained  from  the  general  equations: 

sin  /  sin  (H  —  w)  =  —  sin  i'  sin  (ft'  —  ft) , 

sin  I  cos  (II  —  w)  =  —  sin  i  cos  i'  +  cos  i  sin  i'  cos  (ft'  —  ft) 

=  cos  i  cos  i'  [  —  tan  i  +  tan  i'  cos  (ft'  —  ft)], 
sin  /  sin  (n'  —  w')  =  —  sin  i  sin  (ft'  —  ft), 
sin/ cos  (II'  —  a/)  =  cos  i  sin  i'  —  sin  i  cos  i'  cos  (ft'  —  ft), 

=  cos  i  cos  i'  [tan  it  —  tan  i  cos  (ft'  —  ft)]. 

When  the  Earth  is  the  disturbing  body,  these  become, 

/  =  i;        n  =  180°  +  co;        n'  =  180°  +  *•'  -  ft; 
and  when  the  Earth  is  the  disturbed  body, 

/  =  i'-        n  =  TT  -  ft';        II'  =  *-'  -  ft'. 

As  i,  i'  and  /  are  always  small,  eight   place  logarithms  were  generally  here  used  to 
insure  the  accuracy  of  n  and  n'. 

The  auxiliaries  k,  k',  K,  K'  and  C  were  then  found  from  the  relations: 

k  sin  (A'  -  n)  =    -  cos  I  sin  n';        k  cos  (K  -  n)  =  cos  n';        k'  sin  (K'  -  n)  =  -  sin  n'; 

A;' cos  (K'-U)  =  cos  /  cos  n';         C  =  o'V2, 

and  their  values  were  tested  by  the  equations, 

tan  .7  =  p  ;        tan  \(K  -  K'  +  90°)  cot  |(A'  +  K'  -  90°  -  211)  =  "!"  (^,  ~  "\ , 
•v  sin  (^11  -p  G) 

sin  ( K-  K')  =  sin  I  tan  /  sin  (K'  -  n)  sin  (K  -  n)  cot  n'. 

The  orbit  of  the  disturbed  planet  being  then  divided  into  2j  parts  in  regard  to 
the  eccentric  anomaly,  the  following  equations  were  applied  to  each  point  of  division, 
of  which  those  marked  with  an  asterisk  are  test  equations  upon  the  sums  of  the 
functions  corresponding  respectively  to  the  odd  and  even  points  of  division  of  the 
orbit.  The  sums  corresponding  to  the  odd  points  are  designated  by  Si,  those  to  the 
even  points  by  S2,  and 

2  =  S,  +  S2. 


54  THE    SECULAR   VARIATIONS    OP   THE    ELEMENTS 

r  sin  v  =  a  cos  ip  sin  E, 
r  cos  v  =  a  (cos  E  —  e), 

r2  =  a2  (1  -  2  e  cos  #  +  e2  cos2  .E), 

(the  last  equation  giving  the  value  of  r2  for  use  in  A,  N,  and  J3.     Since 

i  log  r2  =  log  T-, 
this  affords  also  an  independent  test  of  r). 

*S,t;  +  180°  =  S2*>;        *Z^  =  Z2r  =  ja. 
A  =  r2  +  2&aYr  cos  (w  +  X)  +  a'2  =  [r  +  ka'e'  cos  (t>  +  X)]2  +  a'2[l  -  fcV2  cos2  (v  + 

(the  second  form  used  with  ZECH'S  tables  in  the  duplication). 

*SiA  =  ^A  =  ja2  +  |jaV  +  j[a'2  -  2kaa'ee'  cos  X] 
J3  sin  e  =  A;'o'  cos  ^>'r  sin  (v  +  K'} 
B  cos  e  =  ka'r  cos  (t>  +  K)  +  a'2e' 
*2iB  sin  e  =  S2B  sin  e  =  —  jk'aa'  cos  ^>'  •  e  sin  ^' 
*2iB  cos  «  =  22B  cos  e  =  j[a'~e'  —  kaa'e  cos  A'] 
g  =  B2C  sin2  e 


To  effect  the  solution  of  the  cubic  equation,  h  and  I  were  found  from  the  equations, 

the  very  convenient  test  equation,  hi  =  B*    -AC,  being  applied  to  each  pair  of  values. 
The  first  approximation  to  G  was  then  obtained  from 

G  =  h  ~  h(h  -I)' 
and  further  approximations  by  successive  applications  of 


G(G  -  iy 

(The  number  of  trials  required  never  exceeded  three.)     G'  and  G"  then  follow  from 
the  equations, 


G'  =       (A  -  C  -  G)  +      (A  -C-  GY  +        ;        G"  = 
and  we  have  for  verification, 

ft    i     rt/  _   rtii   _      A  ri  .  fir   _    L   _i_  _  a         _  .  fin  _  £ 

^'-''  =  " 


. 

G")' 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS.  55 

(In  some  cases  the  first  approximations  to  G  were  found  by, 

sin  e'  =  -3;        G  =  2q  sin  (60°  -  &')  +  p, 

the  solution  being  then  finished  as  before). 

The  modulus,  (c  =  sin  6),  of  the  elliptic  integrals  employed  in  the  computation 
was  separately  found  by  the  two  equations, 

C*T     I     /""'  /""     I     firr 

sin2  6  =  Q  ,  Q,,  ;        tan2  6  =   Q  _  Q>  , 

and  with  6  as  an  argument  the  values  of  log  K0,  log  L0',  and  log  N0  were  taken  from 
the  tables  of  HILL'S  memoir (8),  the  interpolation  being  effected  in  both  directions  to 
second  differences  by  the  well-known  formulas, 


in  which  n  +  n'  =  1. 

The  logarithms  of  Af,  P,  Q,  and  F  were  then  obtained  from, 


o  . 

"(G  +  G")3'  ~(G  +  (?")2'        y  "(?  +  £"'  W 

the  first  three  being  verified  by  similar  operations  performed  upon  the  values  of  2t 
and  22  formed  from  the  respective  logarithms,  and  the  last  by  the  use  of  ZECH'S 
tables  and  also  by  the  equation, 


V  =  ar-(G  +  G'Tl[GN0  +  G"(N0  -  L0')]K0. 
The  following  auxiliaries  were  next  obtained  : 

Ji'  =  a'2  cos2  <p'[l  -  sin2  /  sin2  (v  +  n)]  +  G" 

=  [a'  cos  >p'  +  a'  cos  <p'  sin  7  sin  (v  +  n)][a'  cos  <p'  —  a'  cos  <?'  sin  /  sin  (v  +  II)]  4-  G", 
J2  =  ka'e'r  sin  (v  +  K)  -  |a'2  cos2  <p'  sin2  1  sin  2(i>  +  n) 

=  ka'e'r  sin  (t;  +  K)  -  a'2  cos2  ^'  sin  (v  +  n)  cos  (v  +  n)  sin2  /, 

the  second  form  being  employed  with  ZECH'S  tables  in  the  duplication 
Jz  =  —  cos2  <p'  sin  I  cos  7  •  r  sin  (t;  +  H)  --  «'  sin  /  sin  II'  •  r2, 

tZ  CL 

a'e' 
*Si/3  =  SaJa  =  -  ja-  cos2  ^'  sin  I  cos  /  •  e  sin  n  ---  sin  /  sin  n'  •  S^2. 

(Z 


=  —  a'2  sin  <p'  cos  <p'  cos  /  •  fi  sin  e, 


56  THE   SECULAR   VARIATIONS   OF   THE   ELEMENTS 

*2iF2  =  S2F2  =  jk'aa'3ee'  cos2  <p'  cos  7  sin  K', 

F3  =  --  sin  49'  cos  y>'  sin  I  •  r  cos  (v  +  II)  •  .B  sin  e. 

There  were  next  obtained, 

B0=    -  N  -QG'  +  7JY;        S0  =  PF*  +  VJ,;        W0  =  PF,  +  VJa; 

fl<">  =  -R0  sin  E;         5<->  =  -S0;        W  =  £.Sfl<">;        £0(c)  =  £. 
r  r  z?  zj 

and  the  very  accurate  test  equation, 

sin  <p  •  !-Ai(>)  +  cos  9?  •  B0(c)  =  0, 
was  applied. 

These  values  were  then  substituted  in  the  following  series  of  equations,  and 
the  final  values  of  the  differential  coefficients  obtained: 


[dc  ~\          TH'TI  1 

~dt  Joo  =  1  +  m  '  COS  *"  '  2?-2fsin  "  '  Ro  +  ^cos  "  +  cos 


m'n      cos  «?    1 


di  1 

sJ0o  = 


mn 


[dftl          m'n 
,. 
a<  Joo      1  + 


sec  <p    1  • 
—    ^-s-.S 
TO     sin  i    2j 


TOrn       If      Or-P1.     ,   .  9v[dx~\         ,   .,iTd«1 
=  ^r-    -  •s-.ZI  —  2-J2p  |  +  2sin17  +2sin2^,    -,.      . 

oo      1  +  TO    2j  a  2  L  rf/  Joo  2  L  rf/  Joo 


<lt 

When  the  Earth  is  the  disturbed  body,  the  third  and  fourth  equations  are  re- 
placed by, 

"  dp  ~|  m'n  1 

-  •  sec  ip--n  -.2  sin  (»  +  «•)•  (r  0 

L  +  TO  2j 

m'n  I  _ 


[da  ~\ 
ZJl,- 

In  this  case 


rdxl  = 

L  dt  Joo 


and  the  last  term  of  the  expression  for  [dL/dt]00  disappears,  but  the  first  two  equations 
remain  unaltered. 

8.  THE  VALUES  OF  THE  PRELIMINARY  CONSTANTS. 

The  values  obtained  for  those  constants  which  are  direct  functions  of  the  ele- 
ments of  the  orbits  in  the  several  cases  are  shown  in  the  following  tables.     The  last 


OF   THE    ORBITS    OF   THE    FOUR   INNER   PLANETS. 


57 


columns  of  these  tables  contain  the  differences  between  the  values  of  K  —  K'  formed 
directly  and  the  same  angles  obtained  from  the  test  formula  of  the  preceding  article. 
The  other  test  equations  were  also  exactly  satisfied. 

An  examination  of  the  formulas  of  the  preceding  article  renders  it  evident  that 
with  any  two  planets  /  will  have  the  same  value  whether  the  inner  or  the  outer 
planet  is  the  disturbing  one,  while  the  value  of  II  in  the  first  case  will  differ  180°  from 
that  of  n'  in  the  second,  and  that  of  n'  in  the  first  case  will  similarly  differ  180° 
from  that  of  II  in  the  second.  These  conditions  will  be  seen  to  be  here  satisfied 
very  exactly,  the  minute  discrepancies  which  occur  being  due  to  the  fact  that  in  some 
places  eight  place  logarithms  were  employed,  in  others  seven,  and  in  still  others  the 
attainment  of  a  higher  accuracy  throughout  the  entire  computation  was  sought  by 
the  use  of  the  dash,  ( — ),  which  was  placed  above  the  last  figure  of  each  logarithm  for 
which  the  interpolation  led  to  a  value  coinciding  more  nearly  with  the  mean  of  the 
two  adjacent  figures  than  with  either  one  of  them.  In  combining  such  logarithms 
the  effect  of  the  dash  was  taken  into  consideration  by  methods  which  are  obvious. 


Mercury  by— 

Venus 

Earth 

Mars 

Jupiter 

Saturn 

Uranus 

Neptune 

Mercury  by — 

Venus 

Earth 

Mars 

Jupiter 

Saturn 

Uranus 

Neptune 

Venus  by — 

Mercury 

Earth 

Mars 

Jupiter 

Saturn 

Uranus 

Neptune 


II 


II' 


K 


4 

7 
5 
6 
6 
6 
7 

20 
0 
9 
17 
23 
19 
1 

42.982 
7.710 
10.165 
15.310 
44.130 
17.399 
42.654 

230 
208 
209 
218 
229 
211 
223 

39 
34 
13 
5 
26 
43 
12 

31.39 
4.99 
54.31 
54.72 
43.69 
10.39 
39.15 

284 
233 
107 
154 
244 
304 
191 

54 
48 
24 
49 
17 
49 
16 

1.27 
31.10 
19.31 
24.01 
50.53 
47.06 
25.42 

305 
334 
101 
63 
345 
266 
324 

43 
57 
45 
8 
17 
43 
34 

2.40 
50.59 
36.16 
32.52 
17.36 
32.95 
3.91 

K' 


log  k 


logfc' 


log<7 


resid. 


305 

47 

57.49 

9.9988328 

9.9999176 

5.3891826 

0.007 

334 

33 

18.85 

9.9978879 

9.9988719 

6.4491252 

0.002 

101 

53 

33.05 

9.9983990 

9.9998432 

8.3052599 

0.004 

63 

24 

30.76 

9.9995281 

9.9978563 

8.7995614 

0.002 

345 

0 

30.90 

9.9978013 

9.9994926 

9.4563012 

0.013 

267 

3 

12.48 

9.9982188 

9.9991396 

9.9089914 

0.000 

324 

24 

7.31 

9.9968502 

9.9998757 

8.8147322 

0.001 

II 


II' 


K 


4 

20 

42.980 

104 

54 

1.27 

50 

39 

31.37 

54 

19 

21.08 

3 

23 

35.010 

234 

7 

49.75 

205 

1 

46.65 

29 

8 

21.75 

1 

56 

2.460 

208 

26 

43.81 

52 

18 

22.07 

156 

9 

18.63 

2 

15 

11.352 

247 

36 

52.56 

130 

2 

45.43 

117 

32 

48.56 

2 

3 

12.046 

281 

7 

33.71 

241 

43 

52.16 

39 

24 

36.81 

2 

37 

16.883 

233 

30 

46.37 

272 

18 

13.25 

321 

12 

24.44 

2 

46 

38.369 

265 

47 

34.23 

179 

34 

46.34 

86 

12 

46.11 

58 

Venus  by  — 

Mercury 
Earth 
Mars 
Jupiter 
Saturn 
Uranus 
Neptune 

Earth  by— 

THE    SECULAR   VARIATIONS   OF 
K'                                   log  k 

54      9     38.85              9.9992531 
29      3    44.28             9.9998637 
156      7    24.89             9.9998450 
117    35    25.70             9.9998033 
39    22    46.29             9.9997836 
321     12    41.79              9.9995460 
86     12     49.67              9.9999999 

/                                      II 

THE    ELEMENTS 
log  k'                           log  C 

9.9994984              7.8017097 
9.9993746              6.4491252 
9.9999075              8.3052599 
9.9998610             8.7995614 
9.9999375              9.4563012 
9.9999992              9.9089914 
9.9994896              8.8147322 

II' 

resid. 

0.000 
0.003 
0.000 
0.001 
0.003 
0.001 
0.001 

K 

Mercury 
Venus 

o 

7 
3 

/ 

0 
23 

7.710 
35.010 

0 

53 

25 

48 
1 

31.10 
46.65 

O 

28 
54 

34 

7 

n 

4.99 
49.75 

0 

25 
330 

i 
25 
56 

13.33 

48.79 

Mars 

1 

51 

2.240 

51 

57 

45.14 

284 

53 

57.15 

127 

3 

21.25 

Jupiter 
Saturn 

1 
2 

18 
29 

42.100 
40.190 

1 

348 

25 
0 

19.94 
50.68 

272 
337 

58 
45 

11.88 
52.32 

88 
10 

27 
13 

5.264 
49.89 

Uranus 

0 

46 

20.540 

27 

7 

31.73 

95 

0 

58.70 

292 

6 

31.40 

Neptune 

1 

47 

1.680 

330 

14 

7.90 

273 

9 

58.47 

57 

4 

3.92 

Earth  by— 


K' 


log  A; 


logfc' 


logC 


resid. 


Mercury 
Venus 

25 
330 

3 
51 

36.28 
5.12 

9.9992608 
9.9994999 

9.9974965 
9.9997387 

7.8017097 
5.3891826 

0.002 
0.008 

Mars 

127 

4 

14.72 

9.9997885 

9.9999850 

8.3052599 

0.009 

Jupiter 
Saturn 

88 
10 

27 
16 

10.859 

6.88 

9.9998865 
9.9999411 

9.9999998 
9.9996473 

8.7995614 
9.4563012 

0.010 
0.003 

Uranus 

292 

6 

34.66 

9.9999609 

9.9999997 

9.9089914 

0.005 

Neptune 

57 

4 

14.94 

9.9997902 

9.9999994 

8.8147322 

0.001 

Mars  by — 


n 


n' 


K 


Mercury 

5 

9 

10.165 

287 

24 

19.31 

29 

13 

54.31 

258 

16 

20.56 

Venus 

1 

56 

2.460 

232 

18 

22.07 

28 

26 

43.81 

203 

52 

27.49 

Earth 

1 

51 

2.240 

104 

53 

57.15 

231 

57 

45.14 

232 

57 

4.23 

Jupiter 

1 

26 

6.381 

149 

47 

4.35 

188 

22 

45.31 

321 

24 

28.37 

Saturn 

2 

21 

52.110 

176 

17 

59.42 

293 

4 

38.76 

243 

12 

17.28 

Uranus 

1 

11 

40.460 

120 

39 

30.31 

315 

36 

26.40 

165 

2 

41.49 

Neptune 

2 

22 

41.388 

152 

49 

56.12 

222 

47 

52.02 

290 

3 

32.71 

Mars  by— 


K' 


log  k 


log   k' 


logt' 


resid. 


Mercury 
Venus 

258 
203 

4 
50 

28.68 
49.03 

9.9995819 
9.9999439 

9.9986621 
9.9998087 

7.8017097 
5.3891826 

0.017 
0.012 

Earth 

232 

55 

19.79 

9.9998596 

9.9999141 

6.4491252 

0.009 

Jupiter 
Saturn 

321 
243 

24 

14 

9.72 
23.99 

9.9999971 
9.9996870 

9.9998667 
9.9999432 

8.7995614 
9.4563012 

0.007 
0.007 

Uranus 

165 

3 

26.31 

9.9999538 

9.9999519 

9.9089914 

0.003 

Neptune 

290 

0 

35.50 

9.9998274 

9.9997986 

8.8147322 

0.006 

OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


59 


9.    THE  RADII  VECTORES  AND  THE  TRUE  ANOMALIES. 

The  values  of  log  r  and  v  for  the  points  of  division  employed  in  the  four  different 
cases  are  given  in  the  following  tables.     In  each  case  the  equations, 

2ir  =  22r  =  ja,     and     2:y  +  180°  =  S2t> 

were  exactly  satisfied,  and  the  values  of  r  were  also  obtained  from  the  equation  stated 
in  Article  7  for  obtaining  the  value  of  r2. 


E 

MEKCUBY. 

log  r 

V 

E 

VENUS. 
logr 

V 

0 

O 

i 

// 

O 

O 

1 

// 

0 

9.4878584 

0 

0 

0.00 

0 

9.8563557 

0 

0 

0.00 

15 

9.4916716 

18 

25 

28.96 

15 

9.8564576 

15 

6 

6.54 

22.5 

9.4963313 

27 

32 

14.93 

30 

9.8567564 

30 

11 

47.87 

30 

9.5026623 

36 

32 

7.50 

45 

9.8572313 

45 

16 

40.52 

45 

9.5195925 

54 

4 

7.02 

60 

9.8578493 

60 

20 

24.50 

60 

9.5407098 

70 

50 

41.41 

75 

9.8585680 

75 

22 

44.64 

67.5 

9.5522314 

78 

55 

7.36 

90 

9.8593378 

90 

23 

31.50 

75 

9.5640735 

86 

46 

40.73 

105 

9.8601064 

105 

22 

42.20 

90 

9.5878217 

101 

51 

53.65 

120 

9.8608213 

120 

20 

20.31 

105 

9.6103385 

116 

9 

54.15 

135 

9.8614342 

135 

16 

35.65 

112.5 

9.6207149 

123 

3 

1.59 

150 

9.8619040 

150 

11 

43.65 

120 

9.6303194 

129 

46 

44.60 

165 

9.8621990 

165 

6 

4.12 

135 

9.6467730 

142 

49 

52.77 

180 

9.8622996 

180 

0 

0.00 

150 

9.6589887 

155 

27 

29.02 

195 

9.8621990 

194 

53 

55.88 

157.5 

9.6633518 

161 

39 

20.97 

210 

9.8619040 

209 

48 

16.35 

165 

9.6664956 

167 

48 

0.75 

225 

9.8614342 

224 

43 

24.35 

180 

9.6690267 

180 

0 

0.00 

240 

9.8608213 

239 

39 

39.69 

195 

9.6664956 

192 

11 

59.25 

255 

9.8601064 

254 

37 

17.80 

202.5 

9.6633518 

198 

20 

39.03 

270 

9.8593378 

269 

36 

28.50 

210 

9.6589887 

204 

32 

30.98 

285 

9.8585680 

284 

37 

15.36 

225 

9.6467730 

217 

10 

7.23 

300 

9.8578493 

299 

39 

35.50 

240 

9.6303194 

230 

13 

15.40 

315 

9.8572313 

314 

43 

19.48 

247.5 

9.6207149 

236 

56 

58.41 

330 

9.8567564 

329 

48 

12.13 

255 

9.6103385 

243 

50 

5.85 

345 

9.8564576 

344 

53 

53.46 

270 

9.5878217 

258 

8 

6.35 

285 

9.5640735 

273 

13 

19.27 

292.5 

9.5522314 

281 

4 

52.64 

300 

9.5407098 

289 

9 

18.59 

315 

9.5195925 

305 

55 

52.98 

330 

9.5026623 

323 

27 

52.50 

337.5 

9.4963313 

332 

27 

45.08 

345 

9.4916716 

341 

34 

31.04 

60 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


E 

THE  EARTH. 
logr 

V 

E 

MARS. 
log  r 

V 

O 

O 

1 

// 

O 

O 

i 

II 

0 

9.9926546 

0 

0 

0.00 

0 

0 

.1403760 

0 

0 

0.00 

22. 

5 

9.9932181 

22 

52 

14.25 

30 

0 

.1463201 

32 

47 

24.62 

30 

9.9936460 

30 

29 

2.39 

45 

0 

.1532670 

48 

54 

53.41 

45 

9.9948189 

45 

41 

0.84 

60 

0 

.1621567 

64 

44 

46.64 

60 

9.9963428 

60 

50 

8.59 

90 

0 

.1828971 

95 

21 

5.913 

67. 

5 

9.9972036 

68 

23 

26.41 

120 

0 

.2026920 

124 

31 

47.15 

90 

0.0000000 

90 

57 

39.46 

135 

0 

.2106341 

138 

39 

52.35 

112. 

5 

0.0027784 

113 

23 

5.92 

150 

0 

.2166313 

152 

34 

23.40 

120 

0.0036266 

120 

49 

43.50 

180 

0 

.2216237 

180 

0 

0.00 

135 

0.0051200 

135 

40 

31.82 

210 

0 

.2166313 

207 

25 

36.60 

150 

0.0062624 

150 

28 

37.29 

225 

0 

.2106341 

221 

20 

7.65 

157.5 

0.0066776 

157 

51 

53.72 

240 

0 

.2026920 

235 

28 

12.85 

180 

0.0072232 

180 

0 

0.00 

270 

0 

.1828971 

264 

38 

54.087 

202. 

5 

0.0066776 

202 

8 

6.29 

300 

0 

.1621567 

295 

15 

13.36 

210 

0.0062624 

209 

31 

22.71 

315 

0.1532670 

311 

5 

6.59 

225 

0.0051200 

224 

19 

28.18 

330 

0 

.1463201 

327 

12 

35.38 

240 

0.0036266 

239 

10 

16.50 

247. 

5 

0.0027784 

246 

36 

54.08 

270 

0.0000000 

269 

2 

20.54 

292. 

5 

9.9972036 

291 

36 

33.59 

300 

9.9963428 

299 

9 

51.41 

315 

9.9948189 

314 

18 

59.16 

330 

9.9936460 

329 

30 

57.61 

337. 

5 

9.9932181 

337 

7 

45.75 

10.     THE   SEPARATE  RESULTS. 

The  values  found  for  the  intermediate  auxiliary  functions  which  depend  upon  £", 
as  well  as  the  final  perturbations  of  the  four  inner  planets  in  each  case  are  now  stated 
in  the  following  tables.  The  results  of  the  application  of  the  more  important  test 
equations  are  also  shown,  but  all  of  the  test  equations  of  Article  7  were  also  applied, 
and  each  computation  (except  the  first),  was,  after  its  completion,  duplicated  from  the 
beginning,  the  forms  of  the  equations  being  changed  in  the  duplication  when  this 
was  possible. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


61 


MERCURY. 

ACTION  OF  VKNUS  ON  MERCURY. 


E 

A 

B  cos  t 

B  sin  e 

1000000  Xff 

h 

0° 

0.619543952 

+  0.13308441 

-  0.18036925 

0.7970904 

0.52358614 

30 

0.627434998 

+  0.22218381 

-  0.06982371 

0.1194506 

0.52390836 

60 

0.647116316 

+  0.24372756 

+  0.07193966 

0.1268000 

0.52384406 

90 

0.675632886 

+  0.19194286 

+  0.20693555 

0.0491867 

0.52344851 

120 

0.706503003 

+  0.08070542 

+  0.29899200 

2.1902889 

0.52319742 

150 

0.730295757 

-  0.06017874 

+  0.32344233 

1.5631633 

0.52358280 

180 

0.738317327 

-  0.19295989 

+  0.27373528 

1.8358797 

0.52446104 

210 

0.727259050 

-  0.28205939 

+  0.16318979 

0.6524819 

0.52500778 

240 

0.701243272 

-  0.30360314 

+  0.02142638 

0.0112481 

0.52470755 

270 

0.669559472 

-  0.25181838 

-  0.11356958 

0.3160138 

0.52391075 

300 

0.641856586 

-  0.14058090 

-  0.20562585 

1.0359483 

0.52329644 

330 

0.624398293 

+  0.00030325 

-  0.23007624 

1.2969588 

0.52323374 

z, 

4.054580456* 

-  0.17962654f 

+  0.28009822J 

5.9972554 

3.14309264 

22 

4.054580456 

-  0.17962659 

+  0.28009814 

5.9972551 

3.14309193 

E 


G 


G' 


G" 


o 

0 

0.09593332 

0.52358258 

0.09595274 

0.000015866 

O 

25 

/ 

20 

n 
53.90 

30 

0.10350215 

0.52390782 

0.10350489 

0.000002203 

26 

23 

25.33 

60 

0.12324776 

0.52384346 

0.12325032 

0.000001964 

29 

0 

59.15 

90 

0.15215988 

0.52344311 

0.15217844 

0.000013171 

32 

37 

46.70 

120 

0.18328109 

0.52318510 

0.18331625 

0.000022837 

36 

17 

45.71 

150 

0.20668846 

0.52356735 

0.20672760 

0.000023681 

38 

55 

52.70 

180 

0.21383179 

0.52444977 

0.21385942 

0.000016369 

39 

41 

12.31 

210 

0.20222678 

0.52500393 

0.20223677 

0.000006145 

38 

21 

51.31 

240 

0.17651123 

0.52470749 

0.17651140 

0.000000121 

35 

27 

1.91 

270 

0.14562423 

0.52390915 

0.14562996 

0.000004142 

31 

49 

7.06 

300 

0.11853565 

0.52329155 

0.11855723 

0.000016698 

28 

25 

30.42 

330 

0.10114005 

0.52322787 

0.10117042 

0.000024501 

26 

5 

20.70 

Si 

0.91134083 

3.14305994 

0.91144736 

0.000073855 

194 

13 

23.40 

S2 

0.91134154 

3.14305922 

0.91144808 

0.000073843 

194 

13 

23.80 

*  6a2  +  3aV  +  6[o'2  -  2kaa'ee'  cos  K]  =  +  4.054580460. 
t  6[a'V  -  kaa'e  cos  K]  =  -  0.17962650. 
t  -  Qk'aa'  cos  <p'-e  sin  K'  =  +  0.28009816. 


62 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION 

OF  VENUS  ON  MERCUBY. 

E 

log*. 

log  Lo' 

log  N0          log  N         log  P 

logQ 

0° 

0.06678154 

0.36107029 

0.27485672     9.0518226     9.9748963 

9.6076810 

30 

0.07267844 

0.36875602 

0.28344481     9.0869397    0.0171823 

9.6511278 

60 

0.08883727 

0.38974368 

0.30686976     9.1792738     0.1306112 

9.7669404 

90 

.0.11429390 

0.42259487 

0.34345542     9.2994384     0.2842725 

9.9240134 

120 

0.14429575 

0.46098687 

0.38608356     9.4147448     0.4383831 

0.0821541 

150 

0.16872258 

0.49199359 

0.42040790    9.4960335    0.5500428 

0.1974492 

180 

0.17620114 

0.50144271 

0.43084933    9.5225000    0.5845077 

0.2336318 

210 

0.16325170 

0.48506821 

0.41274966    9.4887994    0.5335323 

0.1813814 

240 

0.13698082 

0.45165804 

0.37573831    9.4055654    0.4173889 

0.0613864 

270 

0.10823963 

0.41480523 

0.33478930     9.2928156     0.2691020 

9.9083455 

300 

0.08503270 

0.38481172 

0.30136861     9.1761377     0.1234343 

9.7587487 

330 

0.07094409 

0.36649704 

0.28092115    9.0860236    0.0150983 

9.6482336 

Si 

0.69812922 

2.54971331 

2.07576629    5.7500443     1.6692215 

9.5105424 

22 

0.69813034 

2.54971496 

2.07576824    5.7500501     1.6692301 

9.5105508 

E 

logV 

J/ 

1000  X  Ji           J3 

1000  X  Fj 

0° 

9.6076650 

0.521404654 

-  2.7049984      -  0.024195167 

+  0.6439191 

30 

9.6511256 

0.520191194 

-  0.6256166      -  0.032354328 

+  0.2492710 

60 

9.7669384 

0.521003862 

+  1.8268277      -  0.030122813 

-  0.2568249 

90 

9.9240003 

0.522559008 

+  2.6401451      -  0.018096342 

-  0.7387609 

120 

0.0821316 

0.523207843 

+  2.0172777     +  0.000503667 

-  1.0674025 

150 

0.1974260 

0.522626872 

+  1.0213969     +  0.020692271 

-  1.1546906 

180 

0.2336159 

0.521405157 

+  0.3980040     +  0.037057786 

-  0.9772364 

210 

0.1813754 

0.520383901 

+  0.3750436     +  0.045213990 

-  0.5825883 

240 

0.0613863 

0.520288911 

+  0.7059590     +  0.042976535 

-  0.0764923 

270 

9.9083414 

0.521364846 

+  0.6938449     +  0.030947096 

+  0.4054437 

300 

9.7587319 

0.522844254 

-  0.4295092     +  0.012350057 

+  0.7340853 

330 

9.6482088 

0.523030858 

-  2.2812182      -  0.007832614 

+  0.8213734 

Si 

9.5104690 

3.130154681* 

+  1.8135608     +  0.038570065 

-  0.9999517 

S2 

9.5104775 

3.130156679 

+  1.8235957     +  0.038570073 

-  0.9999517 

*  2i(J,'  -  G")  =  +  3.130080826. 
2t(Ji  -  G")  =  +  3.130082836. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


63 


E 
0° 
30 
60 
90 
120 
150 
180 
210 
240 
270 
300 
330 

2! 

22 

ACTION  OF  VENUS  ON  MERCUKY. 
10000  X  F3        #o        1000  X  -So         W,             Rw       1000  X  S<"> 
-  0.24640136   0.05971623   -  0.4883004   -  0.009827036     0.00000000   -  1.5879204 
-  0.00762098   0.06444673   -  0.0208471   -  0.014490447   +  0.10127648   -  0.0655216 
-  0.09149188   0.08146579   +  0.7212240   -  0.017625381   +  0.20314033   +  2.0766362 
-  0.49802851   0.11164771   +  0.7947010   -  0.015287000   +  0.28842180   +  2.0529668 
-  0.89441990   0.15077545   -  0.4917115   +  0.000363093   +  0.30587345   -  1.1518366 
-  0.92808606   0.18433784   -  2.4881611   +  0.032271915   +  0.20211372   -  5.4561938 
-  0.56751805   0.19359808   -  3.0725539   +  0.063241297    0.00000000   -  6.5837414 
-  0.13320998   0.17488253   -  1.4207354   +  0.068605698   -  0.19174661   -  3.1154764 
+  0.01209827   0.14154301   +  0.6131517   +  0.049504580   -  0.28714394   +  1.4363109 
-  0.19301611   0.10799082   +  1.3152406   +  0.025022994   -  0.27897487   +  3.3976883 
-  0.46971306   0.08194590   +  0.7289596   +  0.007023584   -  0.20433749   +  2.0989094 
-  0.49748432   0.06575439   -  0.1643583   -  0.003535808   -  0.10333144   -  0.5165700 

-  2.25744598   0.70904446   -  1.9892305   +  0.092680137   +  0.01753235   -  3.7116419 
-  2.25744596   0.70906002   -  1.9841603   +  0.092587352   +  0.01775908   -  3.7031067 

—  Ro  COS  V 

E 

RO  sin  v 

+  (cos  v  +  cos  E)So 

+  (  -sec2  <f  +  l  )  sin  v  • 
\a       ) 

„    W0  cos  u 

Wo  sin  u 

-2  -Bo 
a 

0° 

-  0.00097660 

-  0.05971623 

-  0.008630594 

-  0.004699311 

-  0.09487655 

30 

+  0.03833160 

-  0.05180530 

-  0.006100201 

-  0.013143840 

-  0.10594280 

60 

+  0.07755258 

-  0.02541165 

+  0.002882594 

-  0.017388064 

-  0.14618182 

90 

+  0.10909895 

+  0.02454507 

+  0.009914505 

-  0.011635936 

-  0.22329541 

120 

+  0.11643405 

+  0.09565744 

-  0.000337472 

+  0.000133975 

-  0.33255092 

150 

+  0.08098459 

+  0.16537954 

-  0.032192272 

-  0.002265848 

-  0.43432170 

180 

+  0.00614511 

+  0.19359808 

-  0.055541660 

-  0.030242129 

-  0.46680548 

210 

-  0.07011629 

+  0.16039915 

-  0.041182679 

-  0.054870101 

-  0.41204381 

240 

-  0.10947718 

+  0.08954949 

-  0.009624818 

-  0.048559933 

-  0.31218788 

270 

-  0.10595403 

+  0.01957233 

+  0.007191939 

-  0.023967193 

-  0.21598164 

300 

-  0.07680517 

-  0.02822239 

+  0.005196776 

-  0.004724857 

-  0.14704334 

330 

-  0.03941900 

-  0.05265111 

-  0.003501678 

+  0.000490095 

-  0.10809245 

2! 

+  0.01287279 

+  0.26545474 

-  0.066055174 

-  0.105480319 

-  1.49964599 

22 

+  0.01292582 

+  0.26543968 

-  0.065870386 

-  0.105392823 

-  1.49967781 

sin  (/>•  |4i(<)  +  cos  <p-BiJ 

c)  =  -  0.0000000083. 

64 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


DIFFERENTIAL  COEFFICIENTS. 


[de/dt]w 
[dx/dt]w 
[di/dt]oQ 


=  +     11321.398  TO' 

=  +1133127.6      TO' 

60449.278  m' 

-  792605.00    TO' 

=  +1127216.0      TO' 

=  -1326653.0      TO' 


log  eoeff. 
p  4.0539001 
p  6.0542788 
n  4.7813911 
n  5.8990568 
p  6.0520072 
n  6. 1227573 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 

[de/dt]m    =  +0.027739414 
[dxAft]oo    =  +2.7763615 
[di/dtlw     =  -0.14811133 
[dfi/dfloo    =  -1.9420214 
[dr/dt]m    =  +2.7618772 
=  -3.2505323 


[dL/di] 


w 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.         Newcomb.  Hill.  Method  of  Gauss. 


[de/dt]m 
e[dw/dt]oo 
[di/dt]w 
GITI  i  r/vo  '/"//i 

bill  I  [Ctlfcy  CttJOO 

[dx/dt]w 

+0.02780 
+0.56811 
-0.14812 
-0.23648 

-3.2769 

+0.02774 
+0.57086 
-0.14806 
-0.23665 

+0.0277391 
+0.567852 
-0.1481112 
-0.2367447 
+2.776347 
-3.250522 

+0.0277394 
+0.567855 
-0.1481113 
-0.2367449 
+2.776361 
-3.250532 

NOTES. 

This  is  the  only  one  of  the  twenty  eight  computations  that  was  not  duplicated, 
but  the  values  of  6  were  computed  by  two  different  formulas  and  all  known  test 
equations  were  applied.  As  an  illustration  of  his  first  modification  of  GAUSS'S 
method  HILL  published  this  complete  computation  from  exactly  the  same  elements 
as  here  employed,  and  DR.  Louis  ARNDT  states  that  he  has  verified  the  results  and 
found  them  correct.  (Bulletin  de  la  Societe  des  Sciences  Naturelles  de  Neuchatel, 
Vol.  XXIV).  INNES  states  however  that  the  test  arising  from  the  constancy  of  the 
major  axis  is  not  satisfied,  the  residual  being  —0.00075  (M.  N.,  Vol.  LII,  page  87), 
but  this  statement  is  an  error,  for  the  residual  obtained  from  HILL'S  figures  is 
—  0.0000000088,  a  practically  exact  agreement  with  that  here  obtained. 


OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS. 


65 


Upon  comparing  the  present  computation  with  that  of  HILL,  the  following 
slight  discrepancies  may  be  noticed : 

n',  K  and  K'  differ  by  less  than  0".l  from  HILL'S  values,  a  difference  doubtless 
due  to  the  fact  that  the  preliminary  computation  was  here  effected  with  eight  place 
logarithms  while  HILL  employed  but  seven.  The  value  of  I  for  330°  should  be 
0.10114009  instead  of  0.11014009,  and  G"  for  180°  should  be  0.00001637  instead  of 
0.00001617.  These  are  misprints  merely.  The  values  of  the  logarithms  of  K0, 
LQ,  and  NQ  in  HILL  seem  to  be  slightly  in  error  throughout,  a  double  interpolation 
to  second  differences  from  HILL'S  values  of  6  giving  with  the  three  functions  most 
in  error, 

Hill's  Values. 

For  E  =  60°,  log  N0  0.30686978  0.3068691 
For  E  =  150°,  log  L0'  0.49199342  0.4919942 
For  E  =  180°,  log  L0'  0.50144261  0.5014421 

The  effect  of  these  differences  upon  the  final  coefficients  is,  however,  almost  in- 
appreciable. 

It  is  evident  from  an  inspection  of  the  final  sums  that  a  division  into  twelve  parts 
is  necessary  in  this  case,  the  terms  from  the  sixth  to  the  eleventh  orders,  inclusive, 
amounting  to  l/600th  of  the  whole  for  [di/dt]00  and  to  l/1200th  of  the  whole  for 


ACTION  OF  THE  EARTH  ON  MERCURY. 

Bcos  € 

+  0.29403604 
+  0.32704393 
+  0.25768623 
+  0.10454706 

-  0.09134000 

-  0.27748723 

-  0.40401673 

-  0.43702459 

-  0.36766690 

-  0.21452774 

-  0.01864062 
+  0.16750659 

-  0.32994198f 

-  0.32994198 

*  6a2  +  3aV  +  6[a'2  -  2kaa'ee'  cos  A]  =  +  6.90363352. 
t  6[a'V  -  kaa'e  cos  A]  =  -  0.3299419S. 
t  -  Qk'aa'  cos  <p'-e  sin  A'  =  +  0.20460788. 


E 

A 

0° 

1.10386215 

30 

1.11164085 

60 

1.12870093 

90 

1.15278960 

120 

1.17861164 

150 

1.19808875 

180 

1.20368356 

210 

1.19273750 

240 

1.16934301 

270 

1.14208711 

300 

1.11943230 

330 

1.10628960 

s, 

6.90363359* 

22 

6.90363341 

B  sin  € 

KXXXTxff 

h 

-  0.13175730 

0.04882864 

1.00008277 

+  0.06103685 

0.01047875 

1.00111148 

+  0.24661358 

0.17106421 

1.00173668 

+  0.37524793 

0.39606073 

1.00128597 

+  0.41247210 

0.47853578 

1.00022588 

+  0.34831240 

0.34124242 

0.99978030 

+  0.19995995 

0.11246339 

1.00066133 

+  0.00716576 

0.00014443 

1.00216106 

-  0.17841099 

0.08952996 

1.00285589 

-  0.30704518 

0.26517354 

1.00212493 

-  0.34426952 

0.33336685 

1.00067710 

-  0.28010968 

0.22068934 

0.99977580 

+  0.20460782t 

1.23378883 

6.00623965 

+  0.20460808 

1.23378921 

6.00623954 

66 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


E 


ACTION  OF  THE  EARTH  ON  MERCURY. 
G  G'  G" 


0 

0 

0.10349811 

1.00007732 

0.10355071      0.00004715 

O 

18 

46'  28.61 

30 

0.11024810 

1.00111031 

0.11025877      0.00000949 

19 

22  58.72 

60 

0.12668299 

1.00171716 

0.12683714      0.00013464 

20 

51  17.40 

90 

0.15122236 

1.00123943 

0.15152995      0.00026105 

22 

54  42.04 

120 

0.17810449 

1.00016768 

0.17843084      0.00026815 

25 

0   4.02 

150 

0.19802718 

0.99973772 

0.19824193      0.00017219 

26 

27   9.31 

180 

0.20274096 

1.00064732 

0.20281046      0.00005541 

26 

45  34.58 

210 

0.19029517 

1.00216104 

0.19029526      0.00000008 

25 

50   0.71 

240 

0.16620585 

1.00284522 

0.16627021      0.00005369 

24 

1  53.32 

270 

0.13968091 

1.00209425 

0.13990074      0.00018915 

21 

57  13.77 

300 

0.11847394 

1.00063933 

0.11879216      0.00028045 

20 

10  34.78 

330 

0.10623253 

0.99975109 

0.10646458      0.00020734 

19 

3  48.08 

2, 

0.89570634 

6.00609403 

0.89669152      0.00083949 

135 

35  52.71 

22 

0.89570625 

6.00609384 

0.89669123      0.00083930 

135 

35  52.63 

E 
0° 

logtfo 

0.03586144 

log  Lo' 
0.32053269 

log  Wo         log  N         log  P 
0.22947519     8.5993188     8.9197434 

logQ 
8.8287400 

30 

0.03828768 

0.32372814 

0.23305754     8.6307049     8.9534609 

8.8632763 

60 

0.04451494 

0.33191845 

0.24223558     8.7125510     9.0428624 

8.9539831 

90 

0.05408514 

0.34447345 

0.25629369     8.8165734     9.1597444 

9.0722159 

120 

0.06487989 

0.35858872 

0.27208270     8.9130565     9.2712667 

9.1849500 

150 

0.07303960 

0.36922627 

0.28397010     8.9788974     9.3482020 

9.2629066 

180 

0.07483693 

0.37156571 

0.28658308     8.0002544     9.3712099 

9.2865324 

210 

0.06949181 

0.36460451 

0.27880654     8.9738846     9.3366140 

9.2517535 

240 

0.05973313 

0.35186484 

0.26456369    8.9063079    9.2556584 

9.1696145 

270 

0.04950056 

0.33846383 

0.24956627    8.8114799    9.1479627 

9.0600557 

300 

0.04158092 

0.32806159 

0.23791429     8.7102233     9.0374863 

8.9477383 

330 

0.03700237 

0.32203558 

0.23116016    8.6301758    8.9522476 

8.8613541 

2i 

0.32140725 

2.06253200 

1.53285453     2.8417119     4.8982271 

4.3715583 

22 

0.32140716 

2.06253178 

1.53285430     2.8417160     4.8982316 

4.3715621 

OF   THE    ORBITS   OF   THE   FOUR   INNER   PLANETS. 


67 


ACTION  OF  THE  EARTH  ON  MERCURY. 


E 
0° 
30 
60 
90 
120 
150 
180 
210 
240 
270 
300 
330 

Si 

iogy 
8.8287147 
8.8632712 
8.9539113 
9.0720772 
9.1848079 
9.2628156 
9.2865032 
9.2517534 
9.1695861 
9.0599551 
8.9475885 
8.8612431 

4.3711117 
4.3711156 

1000  X  F, 

J,'         J2          J, 

0.9963685      -  0.0084115617      -  0.04554861 
0.9875038      -  0.0046146363      -  0.08975347 
0.9853935     +  0.0065532595      -  0.10655243 
0.9913722      +  0.0136249408      -  0.09143424 
0.9979641      +  0.0119853300      -  0.04844492 
0.9998176      +  0.0047535770      +  0.01089164 
0.9963768      -  0.0029434883      +  0.07066640 
0.9902153      -  0.0070671940      +  0.11485779 
0.9854773      -  0.0058653024      +  0.13162975 
0.9862780      -  0.0010764519      +  0.11649805 
0.9932756      +  0.0016290397      +  0.07352222 
0.9996406      -  0.0026304325      +  0.01421265 

5.9548558*     +  0.0029472768      +  0.08527241 
5.9548275      +  0.0029898031      +  0.08527242 

Ko        1000  X  So      1000  X  TF«         «<"> 

1000  XPj 
+  2.1929308 
-  1.0158793 
-  4.1045657 
-  6.2455186 
-  6.8650672 
-  5.7972107 
-  3.3280756 
-  0.1192649 
+  2.9694212 
+  5.1103718 
+  5.7299237 
+  4.6620645 

-  3.4054328 
-  3.4054372 

1000  X  <S("> 

0° 

-  0.18791333 

0.020434768 

-  0.3847186 

-  3.085993 

0.000000000 

-  1.2510793 

30 

+  0.04317409 

0.021303927 

-  0.4280932 

6.547345 

+  0.033478638 

-  1.3454762 

60 

-  0.07397385 

0.025621038 

+  0.1363110 

9.590571 

+  0.063887706 

+  0.3925293 

90 

-  0.49750230 

0.033590134 

+  0.7062450 

-  10.865960 

+  0.086774000 

+  1.8244563 

120 

-  0.86425140 

0.043556201 

+  0.5521780 

7.575463 

+  0.088361000 

+  1.2934800 

150 

-  0.83674077 

0.051546760 

-  0.4218487 

+  1.808294 

+  0.056517506 

-  0.9250562 

180 

-  0.43280690 

0.053431000 

-  1.3516860 

+  13.566615 

0.000000000 

-  2.8963373 

210 

-  0.01035886 

0.048659649 

-  1.2877184 

+  20.505307 

-  0.053351926 

-  2.8237889 

240 

-  0.07819780 

0.040457621 

-  0.3317444 

+  19.465045 

-  0.082075038 

-  0.7771130 

270 

+  0.18040025 

0.032377496 

+  0.5949004 

+  13.349022 

-  0.083641596 

+  1.5368184 

300 

-  0.46718624 

0.026190705 

+  0.7690320 

+  6.465459 

-  0.065308303 

+  2.2142904 

330 

-  0.46610559 

0.022213151 

+  0.2265587 

+  0.990810 

-  0.034907444 

+  0.7120630 

2i 

-  1.94793392 

0.209691333 

-  0.6106280 

+  19.245092 

+  0.004865365 

-  1.0242299 

22 

-  1.94793368 

0.209691117 

-  0.6099562 

+  19.240128 

+  0.004869178 

-  1.0209836 

*  2,(J,'  -  G")  = 

5.9540163. 

2,  (j  '  G")  = 

5.9539882. 

68 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  THE  EARTH  ON  MERCURY. 

•  COS  V  •  Ro 


E 

BUI  v  •  m> 
+  (cos  v  +  cos  E)S<> 

.   (r              ,.\j      „     100  X  TFo  cos  «            100  X  Wo  sin  u 
+  I  -  sec*  if  +  1  1  sin  t'So 

0° 

-  0.000769437 

-  0.020434774           -  0.2710273           -  0.1475729 

30 

+  0.011967951 

-  0.017591057           -  0.2756307           -  0.5938896 

60 

+  0.024315363 

-  0.008157573           -  0.1568518           -  0.9461439 

90 

+  0.032727249 

+  0.008319102           +  0.7047203           -  0.8270793 

120 

+  0.032844271 

+  0.028781472           +  0.7040911            -  0.2795215 

150 

+  0.022159505 

+  0.046499174           -  0.1803831            -  0.0126962 

180 

+  0.002703372 

+  0.053431012            -  1.1914874           -  0.6487582 

210 

-  0.017924683 

+  0.045456375            -  1.2308941            -  1.6399928 

240 

-  0.030714227 

+  0.026434445           -  0.3784440            -  1.9093573 

270 

-  0.031808132 

+  0.005466897           +  0.3836684            -  1.2785779 

300 

-  0.024103783 

-  0.010000896           +  0.4783818           -  0.4349396 

330 

-  0.012845686 

-  0.018098650           +  0.0981246           -  0.0137335 

Si 

+  0.004275559 

+  0.070053686           -  0.5016340           -  4.3662934 

22 

+  0.004276204 

+  0.070051841            -  0.5003946           -  4.3659693 

i 

in  ,.M.">  +  COB  „•*.'• 

>  =  -  0.00000000016. 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 
// 

[de/dt}^    =      +3752.8345  TO'     p  3.5743594 

[dx/dt]M    =  +299037.72      m'    p  5.4757260 

[dildt]M    =  -4591.3713      m'    n  3.6619424 

[dB/<ft]M    =  -328217.95      TO'     n  5.5161623 

[d7r/d*]oo    =  +296589.74      TO'    p  5.4721561 

[dL/dt]m   =  -390282.17      TO'    n  5.5913787 

FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO' 

// 

[d€/dt]w    =  +0.011476557 

[dx/dt]m    =  +0.91448833 

[dildt]<n    =  -0.014040890 

[dQ/dt]oo    =  -1.0037245 

[d7r/di]oo    =  +0.90700208 

[dLldtln   =  -1.1935233 

0.03246657 
0.03502118 
0.04597425 
0.06718020 
0.09606764 
0.12145034 
0.12883338 
0.11464781 
0.08923343 
0.06475510 
0.04699652 
0.03651581 

0.43957179 
0.43957044 


OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS. 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

[de/dtlw    +0.01153  +0.01147  +0.0114766 

e[dr/dt]M   +0.18668  +0.18799  +0.186484 

[di/dtlao    -0.01414  -0.01404  -0.0140409 

sin  i  [dfl/di]oo    -0.12219  -0.12233  -0.122360 

-1.1942  -1.19352 


NOTES. 

As  a'  and  e'  are  both  small  in  this  case,  the  sums,  up  to  and  including  .R0,  are  in 
very  exact  agreement.  But  as  /  and  e  are  unusually  large,  the  final  sums  differ 
considerably,  the  greatest  discrepancy  being  in  W0  cos  u,  which  shows  that  a  neglect 
of  the  terms  from  the  6th  to  the  llth  orders  would  produce  an  error  in  [di/dt]QO  of 
slightly  more  than  1 /1000th  of  the  whole  value  of  this  coefficient. 


70 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  MARS  ON  MERCURY. 


E 

A 

B  cos  £ 

B  sin  e 

<7 

h 

0° 

2.3984504 

+  0.12138918 

+  0.45632916 

0.0042055 

2.3024514 

15 

2.3737047 

-  0.02025060 

+  0.40611421 

0.0033309 

2.3025214 

30 

2.3556032 

-  0.14407564 

+  0.32017434 

0.0020703 

2.3025321 

45 

2.3456111 

-  0.24164776 

+  0.20436670 

0.0008435 

2.3024889 

60 

2.3447258 

-  0.30631712 

+  0.06658267 

0.0000896 

2.3024184 

75 

2.3533241 

-  0.33367707 

-  0.08378742 

0.0001418 

2.3023625 

90 

2.3710514 

-  0.32186246 

-  0.23649650 

0.0011296 

2.3023658 

105 

2.3967846 

-  0.27167903 

-  0.38113729 

0.0029337 

2.3024640 

120 

2.4286852 

-  0.18654648 

-  0.50785318 

0.0052088 

2.3026710 

135 

2.4643476 

-  0.07226635 

-  0.60800831 

0.0074658 

2.3029751 

150 

2.5010251 

+  0.06337314 

-  0.67477750 

0.0091956 

2.3033343 

165 

2.5359018 

+  0.21112855 

-  0.70361065 

0.0099983 

2.3036852 

180 

2.5663693 

+  0.36093056 

-  0.69254258 

0.0096862 

2.3039570 

195 

2.5902662 

+  0.50257031 

-  0.64232765 

0.0083325 

2.3040870 

210 

2.6060491 

+  0.62639549 

-  0.55638795 

0.0062519 

2.3040409 

225 

2.6128739 

+  0.72396747 

-  0.44058010 

0.0039202 

2.3038234 

240 

2.6105920 

+  0.78863682 

-  0.30279611 

0.0018517 

2.3034773 

255 

2.5996753 

+  0.81599664 

-  0.15242602 

0.0004692 

2.3030742 

270 

2.5810993 

+  0.80418226 

+  0.00028298 

0.0000000 

2.3026229 

285 

2.5562149 

+  0.76399902 

+  0.14492387 

0.0004242 

2.3023962 

300 

2.5266327 

+  0.66886612 

+  0.27163969 

0.0014902 

2.3022243 

315 

2.4941374 

+  0.55458616 

+  0.37179487 

0.0027917 

2.3021798 

330 

2.4606272 

+  0.41894655 

+  0.43856414 

0.0038844 

2.3022350 

345 

2.4280691 

+  0.27119119 

+  0.46739720 

0.0044120 

2.3023425 

2, 

29.7509107* 

+  2.89391842f 

-  1.41728084J 

0.0450638 

27.6343997 

22 

29.7509107 

+  2.89391853 

-  1.41728059 

0.0450638 

27.6344002 

*  12a2  +  6aV  +  12[a'2  - 
t  12[a'V  -  kaa'e  COB  K\ 
J  —  12fc'aa'  cos  <p'  •  e  sin 


-  Zkaa'ee'  cos  A')  =  29.7509106. 
=  +  2.89391844. 
A"'  =  -  1.41728062. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


71 


E 


ACTION  OF  MARS  ON  MERCURY. 
G  G' 


0 

0.0758032 

2.3016305 

0.0957141 

0.0190900 

12 

51 

3.65 

15 

0.0509875 

2.3018785 

0.0717874 

0.0201570 

11 

28 

40.15 

30 

0.0328754 

2.3021359 

0.0509293 

0.0176577 

9 

54 

3.88 

45 

0.0229265 

2.3023282 

0.0338958 

0.0108086 

7 

59 

27.93 

60 

0.0221116 

2.3024014 

0.0237650 

0.0016363 

6 

1 

37.49 

75 

0.0307659 

2.3023354 

0.0326775 

0.0018845 

7 

2 

5.28 

90 

0.0484898 

2.3021481 

0.0572743 

0.0085668 

9 

43 

5.24 

105 

0.0741248 

2.3018919 

0.0890147 

0.0143178 

12 

11 

37.36 

120 

0.1058185 

2.3016404 

0.1249596 

0.0181104 

14 

22 

46.33 

135 

0.1411768 

2.3014735 

0.1626257 

0.0199473 

16 

17 

11.47 

150 

0.1774951 

2.3014531 

0.1994129 

0.0200367 

17 

54 

21.70 

165 

0.2120208 

2.3016063 

0.2327627 

0.0186630 

19 

13 

8.83 

180 

0.2422166 

2.3019140 

0.2604177 

0.0161582 

20 

12 

26.11 

195 

0.2659835 

2.3023097 

0.2806562 

0.0128954 

20 

51 

34.72 

210 

0.2818125 

2.3026974 

0.2924401 

0.0092842 

21 

10 

38.08 

225 

0.2888548 

2.3029783 

0.2954612 

0.0057613 

21 

10 

27.64 

240 

0.2869190 

2.3030785 

0.2900893 

0.0027715 

20 

52 

41.30 

255 

0.2764054 

2.3029737 

0.2772408 

0.0007349 

20 

19 

35.05 

270 

0.2582114 

2.3026922 

0.2582114 

0.0000000 

19 

33 

51.59 

285 

0.2336230 

2.3023071 

0.2344977 

0.0007857 

18 

38 

24.76 

300 

0.2042127 

2.3019157 

0.2076391 

0.0031178 

17 

36 

2.08 

315 

0.1717618 

2.3016103 

0.1791035 

0.0067722 

16 

29 

5.95 

330 

0.1381965 

2.3014548 

0.1502129 

0.0112362 

15 

19 

13.32 

345 

0.1055309 

2.3014695 

0.1221038 

0.0156999 

14 

6 

53.42 

2, 

1.8741623 

27.6251620 

2.0110657 

0.1276658 

185 

31 

50.77 

22 

1.8741617 

27.6251624 

2.0118270 

0.1284276 

185 

48 

12.56 

72 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  MARS  ON 

MERCURY. 

E 

logtfo 

log  LJ 

logtfo 

log  JV 

logP 

logQ 

0° 

0.01657774 

0.29504379 

0.20087027 

8.0316820 

7.5954802 

7.8669295 

15 

0.01319292 

0.29055307 

0.19582511 

8.0355546 

7.5943701 

7.8655109 

30- 

0.00979383 

0.28603832 

0.19075139 

8.0547661 

7.6099058 

7.8800682 

45 

0.00636436 

0.28147805 

0.18562486 

8.0870691 

7.6401445 

7.9084927 

60 

0.00361341 

0.27781623 

0.18150720 

8.1291203 

7.6819573 

7.9481379 

75 

0.00492729 

0.27956556 

0.18347444 

8.1771100 

7.7316276 

7.9980604 

90 

0.00943284 

0.28555854 

0.19021211 

8.2272783 

7.7853440 

8.0537440 

105 

0.01490785 

0.29282894 

0.19838217 

8.2762397 

7.8395128 

8.1098440 

120 

0.02081742 

0.30066160 

0.20717937 

8.3211159 

7.8908949 

8.1628540 

135 

0.02681707 

0.30859802 

0.21608807 

8.3595540 

7.9366442 

8.2098882 

150 

0.03254501 

0.31616047 

0.22457224 

8.3896939 

7.9743208 

8.2484993 

165 

0.03762579 

0.32285663 

0.23208058 

8.4101312 

8.0019112 

8.2766735 

180 

0.04171228 

0.32823434 

0.23810787 

8.4198969 

8.0178772 

8.2928778 

195 

0.04453613 

0.33194630 

0.24226678 

8.4184648 

8.0212321 

8.2961421 

210 

0.04594748 

0.33380024 

0.24434354 

8.4057702 

8.0116018 

8.2861294 

225 

0.04593448 

0.33378318 

0.24432441 

8.3822399 

7.9892733 

8.2631894 

240 

0.04461766 

0.33205341 

0.24238678 

8.3488317 

7.9552231 

8.2283§75 

255 

0.04222050 

0.32890264 

0.23885673 

8.3070780 

7.9111256 

8.1835072 

270 

0.03902753 

0.32470208 

0.23414923 

8.2591387 

7.8593690 

8.1310520 

285 

0.03533705 

0.31984157 

0.22870029 

8.2078386 

7.8030574 

8.0742275 

300 

0.03141955 

0.31467566 

0.22290683 

8.15CG450 

7.7459663 

8.0168746 

315 

0.02748746 

0.30948385 

0.21708212 

8.1095325 

7.6924008 

7.9633068 

330 

0.02367787 

0.30444739 

0.21142959 

8.0706478 

7.6468600 

7.9179598 

345 

0.02004728 

0.29964170 

0.20603415 

8.0437755 

7.6135016 

7.8848519 

Si 

0.31918262 

3.69919207 

2.58841642 

8.8145868 

3.7748004 

7.0335140 

ss 

0.31939818 

3.69947951 

2.58873971 

8.8145879 

3.7748012 

7.0338946 

OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


73 


ACTION  OF  MARS  ON  MERCURY. 


E 

logf 

/»' 

Ji 

0° 

7.8624692 

2.3161036 

+  0.034714816 

15 

7.8607965 

2.3114562 

+  0.028725995 

30 

7.8759318 

•  2.3036617 

+  0.023022893 

45 

7.9059548 

2.2939379 

+  0.017028648 

60 

7.9477527 

2.2850812 

+  0.009527241 

75 

7.9976170 

2.2883310 

-  0.000387926 

90 

8.0517340 

2.2994643 

-  0.012768514 

105 

8.1064943 

2.3097716 

-  0.026802369 

120 

8.1586286 

2.3171693 

-  0.041120811 

135 

8.2052458 

2.3210340 

-  0.054165965 

150 

8.2438461 

2.3213532 

-  0.064480743 

165 

8.2723467 

2.3185110 

-  0.070890916 

180 

8.2891363 

2.3131717 

-  0.072601961 

195 

8.2931581 

2.3062073 

-  0.069243148 

210 

8.2839812 

2.2986444 

-  0.060886015 

225 

8.2618557 

2.2916136 

-  0.048054879 

240 

8.2277453 

2.2862701 

-  0.031734816 

255 

8.1833367 

2.2836646 

-  0.013368882 

270 

8.1310520 

2.2845306 

+  0.005188260 

285 

8.0740447 

2.2890080 

+  0.021821056 

300 

8.0161484 

2.2963713 

+  0.034481515 

315 

7.9617277 

2.3049376 

+  0.041716329 

330 

7.9153379 

2.3123762 

+  0.043244020 

345 

7.8811861 

2.3164879 

+  0.040207969 

S, 

7.0037635 

27.6341976* 

-  0.133414115 

22 

7.0037641 

27.6349607 

-  0.133414088 

*Si(Ji'  -G")  =27, 

5065318. 

22(-/i'  -  G")  =  27 

,5065331. 

-  0.08283714 

-  0.12496689 

-  0.15748255 

-  0.17817560 

-  0.18564570 

-  0.17939386 

-  0.15985316 

-  0.12835823 

-  0.08705254 

-  0.03874380 
+  0.01328579 
+  0.06550049 
+  0.11434922 
+  0.15650567 
+  0.18909426 
+  0.20988694 
+  0.21745674 
+  0.21127772 
+  0.19176389 
+  0.16024221 
+  0.11886354 
+  0.07045517 
+  0.01832594 

-  0.03396170 

+  0.19026829 
+  0.19026812 


-  0.09798270 

-  0.08720058 

-  0.06874762 

-  0.04388148 

-  0.01429659 
+  0.01799079 
+  0.05078038 
+  0.08183755 
+  0.10904591 
+  0.13055114 
+  0.14488783 
+  0.15107882 
+  0.14870229 
+  0.13792019 
+  0.11946725 
+  0.09460109 
+  0.06501619 
+  0.03272882 

-  0.00006076 

-  0.03111796 

-  0.05832630 

-  0.07983156 

-  0.09416821 

-  0.10035921 

+  0.30431767 
+  0.30431761 


74 


THE    SECULAE   VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  MARS  ON  MERCURY. 


E 

ft 

#0 

1000  X  So 

100  X  Wv 

fl<»> 

0° 

+  0.006125234 

0.005413043 

-  0.1331172 

-  0.05793911 

0.000000000 

15 

+  0.004244733 

0.005395946 

-  0.1341971 

-  0.08902862 

+  0.004501863 

30 

+  0.002091460 

0.005581735 

-  0.1069853 

-  0.11749705 

4-  0.008771561 

45 

+  0.000394535 

0.005978460 

-  0.0544826 

-  0.14331159 

+  0.012778556 

60 

-  0.000202379 

0.006587462 

+  0.0157366 

-  0.16470025 

4-  0.016426268 

75 

+  0.000673582 

0.007397547 

+  0.0931210 

-  0.17804909 

4-  0.019496561 

90 

+  0.003010065 

0.008379102 

+  0.1659322 

-  0.17823953 

4-  0.021645905 

105 

4-  0.006397644 

0.009479745 

+  0.2230373 

-  0.15960687 

+  0.022459598 

120 

+  0.010124788 

0.010622853 

+  0.2557091 

-  0.11755697 

4-  0.021550254 

135 

+  0.013355105 

0.011710967 

+  0.2593958 

-  0.05060872 

4-  0.018676901 

150 

+  0.015340435 

0.012635664 

+  0.2351775 

+  0.03775311 

4-  0.013854131 

165 

+  0.015613320 

0.013293529 

+  0.1902485 

+  0.13831063 

4-  0.007415508 

180 

+  0.014107795 

0.013605711 

4-  0.1366988 

+  0.23722085 

0.000000000 

195 

+  0.011176655 

0.013535474 

+  0.0883172 

+  0.31912559 

-  0.007550472 

210 

+  0.007500348 

0.013096767 

+  0.0561741 

+  0.37133333 

-  0.014359702 

225 

+  0.003911780 

0.012350565 

+  0.0447317 

4-  0.38738325 

-  0.019696946 

240 

+  0.001183526 

0.011390146 

+  0.0503246 

+  0.36844987 

-  0.023106835 

255 

-  0.000166297 

0.010320547 

+  0.0628133 

+  0.32211263 

-  0.024451640 

270 

+  0.000001635 

0.009239635 

+  0.0697179 

+  0.25931088 

-  0.023868989 

285 

+  0.001425688 

0.008225494 

+  0.0610457 

+  0.19093564 

-  0.021678655 

300 

+  0.003528129 

0.007331827 

+  0.0329172 

4-  0.12533212 

-  0.018282391 

315 

+  0.005583303 

0.006590510 

-  0.0111915 

+  0.06726176 

-  0.014086776 

330 

+  0.006923128 

0.006018069 

-  0.0617561 

4-  0.01815027 

-  0.009457250 

345 

+  0.007124120 

0.005623237 

-  0.1063107 

-  0.02290733 

-  0.004691492 

Si 

+  0.069734164 

0.109902014 

+  0.7165294 

4-  0.78161752 

-  0.006827048 

s2 

4-  0.069734168 

0.109902021 

+  0.7165286 

+  0.78161728 

-  0.006826894 

i">  +  cos  <f  •  B0<c)  =  +  0.000000000104. 


OF   THE    DEBITS   OF   THE    FOUR   INNER   PLANETS. 


75 


ACTION  OF  MARS  ON  MERCURY. 


1000x-.RoCosB+ 


E 

1000  X  S<" 

+  «B  '      "    -  10°°  X  W"  c°s  u  10°°  X  W°  8in  « 

-2-#o 

C  ' 

(  -  secV  +  1  1  sin  vSf, 

a 

0° 

-0.43288822 

-0.2662344 

-  5.4130432 

-0.5088501    -  0.2770661 

-0.008600192 

15 

-0.43258505 

+  1.4484906 

-  5.1972533 

-0.6072560    -  0.6510373 

-0.008648634 

30 

-0.33624961 

+3.1443027 

-  4.6032128 

-0.4946402    -  1.0657799 

-0.009175713 

45 

-0.16468928 

+4.7703837 

-  3.5917373 

-0.1836694    -  1.4212974 

-0.010218569 

60 

+0.04531053 

+6.2357682 

-  2.1327368 

+0.2693639    -  1.6248266 

-0.011820509 

75 

+0.25408211 

+7.4151862 

-  0.2308954 

+0.7622000    -  1.6090993 

-0.014007787 

90 

+0.42865608 

+8.1659686 

+  2.0547246 

+  1.1559864    -  1.3566975 

-0.016758205 

105 

+0.54706506 

+  8.3522568 

+  4.6004959 

+  1.3031443    -  0.9215481 

-0.019968410 

120 

+  0.59900000 

+7.8723845 

+  7.2196098 

+  1.0926174    -  0.4337649 

-0.023429816 

135 

+0.58504581 

+6.6852166 

+  9.6761487 

+0.5003962    -  0.0756834 

-0.026827124 

150 

+0.51571179 

+4.8307371 

+  11.7119607 

-0.3765995    -  0.0265069 

-0.029771109 

165 

+0.41003953 

+2.4394871 

+  13.0838349 

-1.3270492    -  0.3897732 

-0.031867221 

180 

+0.29291257 

-0.2733976 

+  13.6057094 

-2.0833919    -  1.1343956 

-0.032806220 

195 

+  0.19034846 

-3.0319641 

+  13.1877783 

-2.4169344    -  2.0838761 

-0.032447216 

210 

+0.12318210 

-5.5396099 

+  11.8615372 

-2.2290420    -  2.9698842 

-0.030857525 

225 

+0.10088867 

-7.5290371 

+  9.7823314 

-1.5918740   -  3.5316447 

-0.028292299 

240 

+  0.11788553 

-8.8108894 

+  7.2045350 

-0.7163503    -  3.6141908 

-0.025122156 

255 

+0.15406833 

-9.3069299 

+  4.4325573 

+0.1350559    -  3.2182941 

-0.021739497 

270 

+0.18010328 

-9.0565656 

+  1.7602448 

+0.7452938    -  2.4836977 

-0.018479268 

285 

+0.16656410 

-8.1932578 

-  0.5835177 

+  1.0058653    -  1.6229225 

-0.015575563 

300 

+0.09477911 

-6.8986293 

-  2.4659985 

+0.9273370    -  0.8431253 

-0.013156194 

315 

-0.03382955 

-5.3509513 

-  3.8502686 

+0.6070918   -  0.2895753 

-0.011264707 

330 

-0.19409650 

-3.6857779 

-  4.7671354 

+0.1797507    -  0.0251579 

-0.009892996 

345 

-0.34269706 

-1.9808199 

-  5.2732726 

-0.2254930    -  0.0403424 

-0.009012936 

2! 

+  1.43430766 

-4.2819430 

+36.0361948 

-2.0385248   -15.8550934 

-0.229869903 

22 

+  1.43430113 

-4.2819391 

+36.0362016 

-  2.0385225   -  15.8550938 

-0.229869963 

DIFFERENTIAL  COEFFICIENTS. 

w 

log  coeff. 

1  fif>  //r/1 

I  U-C'  /  Ltt-lQQ 

1879.077 

TO'  n  3.2739445 

[dxMJoo 

=  +76914.75 

TO'  p  4.8860096 

[dildt]m 

934.0667 

TO'  n  2.9703779 

[dn/dt]m 

=  -59594.26 

TO'  n  4.7752044 

[dTr/dt}m 

=  +76470.27 

TO'  p  4.8834926 

[dL/dt]w 

=  -101879.0 

TO'  n  5.0080846 

76  THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUES  OF  m'. 


=  -0.00060742746 
[dx/d4»    =  +0.024863343 
[dt'/dfloo     =  -0.00030194497 
[dQ/dfloo    =  -0.019264347 
[drfdt]m    =  +0.024719659 

=  -0.032933242 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


[de/dtlw 

-0.00060 

-0.00061 

-0.000607 

< 

>,[dir/dt]oo 

+0.00508 

+0.00511 

+0.005082 

[dt'/dflw 

-0.00030 

-0.00030 

-0.000302 

sin  i 

[dQ/d*]oo 

-0.00234 

-0.00235 

-0.002348 

[dL/d/]oo 

-0.0331 

-0.032933 

NOTES. 

On  account  of  the  very  large  values  of  the  eccentricities  of  both  orbits  and  their 
high  mutual  inclination,  the  approximate  test  is  here  wholly  inapplicable  if  but 
twelve  points  of  division  are  employed.  Thus  the  two  sums  differ  by  1°  38'  46".  90 
for  6  and  by  40'  42".  47  for  «,  while  the  sums  of  the  functions  immediately  dependent 
upon  these  quantities  differ  by  proportionate  amounts.  When  the  number  of  points 
of  division  is  increased  to  twenty-four,  the  final  sums  are  in  almost  exact  agreement, 
showing  that  the  combined  effect  of  all  terms  from  the  llth  to  the  23rd  orders  is 
wholly  inappreciable.  The  greatest  difference  which  arises  in  the  variations  from 
the  employment  of  twenty-four  points  of  division,  instead  of  twelve,  occurs  in  the 
case  of  [di/dt]00  and  here  produces  a  decrease  of  but  three  units  in  the  seventh  decimal 
of  the  logarithm  of  the  coefficient, 


OF  THE  ORBITS  OF  THE  FOUK  INNER  PLANETS. 


77 


E 

0° 

30 

60 

90 

120 

150 

180 

210 

240 

270 

300 

330 


0 

30 

60 

90 

120 

150 

180 

210 

240 

270 

300 

330 


ACTION  OF  JUPITER  ON  MERCURY. 

A 

B  cos  e 

Bam  f 

p 

27.23340536 

+2.0282403 

+  1.4219711 

9.05679111 

27.14356714 

+  1.0282450 

+  1.6206004 

9.02684503 

27.06879996 

+0.0526625 

+  1.2863778 

9.00192264 

27.03145602 

-0.6371012 

+0.5088565 

8.98947466 

27.04270097 

-0.8562240 

-0.5036265 

8.99322298 

27.09836241 

-0.5459922 

-1.4797775 

9.01177679 

27.18120744 

+0.2104677 

-2.1580381 

9.03939180 

27.26787830 

+  1.2104630 

-2.3566674 

9.06828209 

27.33631108 

+2.1860457 

-2.0224444 

9.09109301 

27.37048776 

+2.8758091 

-1.2449232 

9.10248524 

27.36240999 

+3.0949315 

-0.2324396 

9.09979265 

27.31308297 

+2.7846997 

+0.7437107 

9.08335031 

163.22483480* 

+6.7161237f 

-2.2081997J 

54.28221419 

163.22483460 

+6.7161234 

-2.2082005 

54.28221412 

-  8' 

9 

h 

I 

0111 

88  49   5 

0.12745094 

27.006742 

+0.163630 

89   2  45 

0.16554389 

27.007491 

+0.073044 

89  19   0 

0.10430339 

27.007569 

-0.001801 

89  39  30 

0.01632120 

27.006899 

-0.038475 

89  41  40 

0.01598743 

27.006246 

-0.026577 

89  11  43 

0.13801114 

27.006455 

+0.028875 

88  41  46 

0.29354831 

27.007536 

+0.110638 

88  20  11 

0.35007250 

27.008599 

+0.196247 

87  48   4 

0.25781881 

27.008698 

+0.264581 

88  19  32 

0.09768920 

27.007728 

+0.299728 

88  22  22 

0.00340551 

27.006562 

+0.292816 

88  35  32 

0.03486337 

27.006176 

+0.243874 

532°  41'  57" 

0.80250439 

162.043353 

+0.803287 

533   9  13 

0.80250130 

162.043348 

+0.803293 

•f 

80.552426 
80.826371 

81.050843 
81.156986 
81.117340 
80.952183 
80.714558 
80.466952 
80.265977 
80.156946 
80.170254 
80.311910 

483.871398 
483.871348 


G 

27.006566 
27.007263 
27.007426 
27.006877 
27.006224 
27.006265 
27.007132 
27.008116 
27.008341 
27.007593 
27.006557 
27.006128 

162.042246 
162.042242 


*  6o2  +  3o%2  +  6[a'2  -  2Jtaa'ee'  cos  K]  =  +  163.22483477. 
t  6[o'2  e'  -  koa'e  cos  A']  =  +  6.7161238. 
J  -  Sk'aa'  cos  »>'  •  e  sin  K'  =  -  2.2082004. 


78 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


G' 


ACTION  OF  JUPITER  ON  MERCURY. 


G" 


log  ZV 


0 

0.188801 

0.024995 

5 

6 

8.07 

0.0025877 

0.2764500 

30 

0.123075 

0.049804 

4 

35 

5.15 

0.0020887 

0.2757852 

60 

0.061321 

0.062980 

3 

53 

7.75 

0.0014995 

0.2750001 

90 

0.011983 

0.050435 

2 

45 

10.71 

0.0007524 

0.2740043 

120 

0.014441 

0.040995 

2 

35 

41.32 

0.0006684 

0.2738923 

150 

0.087484 

0.058419 

4 

12 

38.17 

0.0017613 

0.2753489 

180 

0.173639 

0.062597 

5 

21 

37.00 

0.0028566 

0.2768082 

210 

0.248823 

0.052092 

6 

3 

11.63 

0.0036449 

0.2778582 

240 

0.297071 

0.032133 

6 

20 

5.30 

0.0039929 

0.2783216 

270 

0.311476 

0.011613 

6 

16 

40.56 

0.0039213 

0.2782263 

300 

0.293250 

0.000430 

5 

59 

8.38 

0.0035638 

0.2777502 

330 

0.249105 

0.005182 

5 

34 

4.63 

0.0030827 

0.2771094 

o 

/ 

// 

Si 

1.028523 

0.224130 

29 

15 

47.82 

0.0151689 

1.6582224 

22 

1.031946 

0.227545 

29 

26 

50.85 

0.0152513 

1.6583323 

E 

logtfo 

log  AT 

logP 

logQ 

logV 

0° 

0.1799707 

6.4183196 

3.8310274 

5.1664192 

5.1659174 

30 

0.1792229 

6.4468140 

3.8580378 

5.1937562 

5.1927566 

60 

0.1783398 

6.5219986 

3.9320091 

5.2678436 

5.2665795 

90 

0.1772196 

6.6157906 

4.0252257 

5.3607256 

5.3597128 

120 

0.1770937 

6.7009450 

4.1105921 

5.4459161 

5.4450927 

150 

0.1787322 

6.7589559 

4.1694988 

5.5052851 

5.5041125 

180 

0.1803735 

6.7800058 

4.1918462 

5.5278954 

5.5266396 

210 

0.1815544 

6.7609474 

4.1741434 

5.5101707 

5.5091259 

240 

0.1820755 

6.7044320 

4.1187250 

5.4544932 

5.4538485 

270 

0.1819683 

6.6198775 

4.0347585 

5.3701732 

5.3699402 

300 

0.1814329 

6.5255910 

3.9403890 

5.2755478 

5.2755392 

330 

0.1807123 

6.4489104 

3.8629284 

5.1980770 

5.1979730 

Si 

1.0792861 

9.6512920 

4.1245888 

2.1381153 

2.1336169 

S2 

1.0794097 

9.6512958 

4.1245926 

2.1381878 

2.1336210 

OF   THE    ORBITS   OF   THE   FOUR   INNER   PLANETS. 


79 


ACTION  OF  JUPITER  ON  MERCUKY. 

E 

JV 

J. 

J3 

Ft 

Vi 

0° 

26.907806 

-0.08848287 

-1.4438584 

-1.8440911 

+0.12702123 

30 

26.754757 

-0.00410757 

-2.3330959 

-2.1016844 

+0.05043578 

60 

26.779325 

+0.16212886 

-2.4984534 

-1.6682462 

-0.05353664 

90 

26.922525 

+0.18465069 

-1.8956909 

-0.6599133 

-0.05566955 

120 

27.032841 

+0.04256694 

-0.6863554 

+0.6531309 

+0.07759141 

150 

27.046752 

-0.14516517 

+0.8055488 

+  1.9190575 

+0.24215469 

180 

26.945408 

-0.26169947 

+2.1803373 

+2.7986639 

+0.29255865 

210 

26.802715 

-0.24648650 

+3.0696706 

+3.0562500 

+0.18231424 

240 

26.714620 

-0.10777763 

+3.2352193 

+  2.6228182 

+0.00934620 

270 

26.757114 

+0.06770763 

+2.6325527 

+  1.6144859 

-0.07863175 

300 

26.911783 

+0.13569580 

+  1.4231210 

+0.3014406 

-0.02506303 

330 

27.011072 

+0.02690450 

-0.0689744 

-0.9644853 

+0.08731406 

Si 

161.291783* 

-0.11756837 

+2.2100104 

+2.8637163 

+0.42791782 

22 

161.294935 

-0.11649642 

+2.2100109 

+2.8637104 

+0.42791747 

E 

1000  X  Ro 

1000,000  X  So 

100,000  X  W, 

1000  X  R(n> 

100,000  X  S<"> 

0° 

0.12949124 

-0.25462237 

-  2.1070330 

0.00000000 

-0.82801500 

30 

0.13531974 

-0.15796961 

-  3.6329106 

+0.21265173 

-0.49649079 

60 

0.16094783 

+0.15688051 

-  4.6204172 

+0.40133398 

+0.45170948 

90 

0.20322837 

+0.35279259 

-  4.3458069 

+0.52500398 

+0.91137646 

120 

0.25064630 

+0.20287634 

-  1.9026678 

+0.50847837 

+0.47523891 

150 

0.28657259 

-0.17989808 

+  2.6073819 

+0.31420703 

-0.39449173 

180 

0.29756923 

-0.44461029 

+  7.3765067 

0.00000000 

-0.95269261 

210 

0.28082350 

-0.33962404 

+  9.9405132 

-0.30790355 

-0.74474862 

240 

0.24483424 

+0.03827880 

+  9.2004732 

-0.49668759 

+0.08966829 

270 

0.20310516 

+0.33360150 

+  6.1619379 

-0.52468566 

+0.86179960 

300 

0.16660105 

+0.28219872 

+  2.6818065 

-0.41543058 

+0.81254093 

330 

0.14104055 

-0.02790130 

-  0.1024400 

-0.22164184 

-0.08769243 

Si 

1.25008989 

-0.01899829 

+  10.6286684 

-0.00230582 

+0.04845000 

S2 

1.25008991 

-0.01899894 

+  10.6286755 

-0.00236831 

+0.04975249 

sin  <f  •  \  4i("'+  cos  if 

•  Bo(c)  =  +  0.00000000000073. 

*2,(Ji'  -G")  =  161.067653. 

S2(Ji'  -  G")  =  161.067390. 

80 


THE    SECULAR   VARIATIONS   OF    THE   ELEMENTS 


ACTION  or  JUPITER  ON  MERCURY. 


E 


1000  X  [ft.  sin  v  + 
(cos  v  +  cos  E)S(,] 


1000  X  Wi,  cos  u  1000  X  Wa  sin  u 


1000  X 


DIFFERENTIAL  COEFFICIENTS. 


log  cqeff. 


[dx/dt]w 
[di/dt]oo 


+       3.3470577  m'    p  0.5246632 
+  1613.8089       m'    p  3.2078521 

51.404941  m' 
- 1550.4039  TO' 
+  1602.2454  TO' 


n  1.7110049 
w  3.1904449 
p  3.2047290 


[dL/dt]w   =  -2312.2863        m'    n  3.3640416 


(-';*) 


0° 

-0.00509245 

-0.12949124 

-0.018505013 

-0.010075877 

-0.20573445 

30 

+0.07792118 

-0.11047546 

-0.015293859 

-0.032953015 

-0.22244969 

60 

+0.15333585 

-0.04994118 

+0.007556597 

-0.045582053 

-0.28880400 

90 

+0.19816106 

+0.04884221 

+0.028185071 

-0.033078773 

-0.40645673 

120 

+0.19031365 

+0.16372520 

+0.017684089 

-0.007020516 

-0.55282671 

150 

+0.12222497 

+0.25901646 

-0.026009473 

-0.001830673 

-0.67519875 

180 

+0.00889221 

+0.29756923 

-0.064784149 

-0.035274618 

-0.71750180 

210 

-0.11061196 

+0.25859897 

-0.059670973 

-0.079503145 

-0.66165318 

240 

-0.18859570 

+0.15601907 

-0.017887815 

-0.090249100 

-0.54000750 

270 

-0.19945173 

+0.03508579 

+0.017710220 

-0.059019459 

-0.40621028 

300 

-0.15503995 

-0.05982940 

+0.019842790 

-0.018040859 

-0.29894808 

330 

-0.08443003 

-0.11301587 

-0.001014512 

+0.000141991 

-0.23185401 

Zi 

+0.00381361 

+0.37805168 

-0.056093501 

-0,206243023 

-2.60382254 

22 

+0.00381349 

+0.37805210 

-0.056093526 

-0.206243074 

-2.60382263 

FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  m'. 

[de/dfloo    =  +0.00319413 

[dx/dt]w    =  +1.540072 

[di/d/]oo     =  -0.049056191 

[dtt/dtlw    =  -1.4795642 

[djr/dflw    =  +1.5290366 

=  -2.2066350 


OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  81 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


[de/dt]m 

+0^00320 

+0.00320 

+0.003194 

e[dw/dt]M 

+0.31437 

+0.31664 

+0.314377 

[di/dt]oo 

-0.04907 

-0.04905 

-0.049056 

sin  i  [dfl/dt]oo 

-0.18042 

-0.18037 

-0.180368 

[dL/dt]M 

-2.2078 

-2.20663 

NOTES. 

The  above  results  were  published  in  1896  in  A.  J.,  No.  386.  In  1911,  upon 
applying  to  the  various  computations  all  of  the  test  equations  devised  or  learned  of 
by  that  time,  a  slight  error  was  detected  in  the  value  of  Fz  for  240°.  This  rendered 
the  values  of  W0,  W0  cos  u,  W0  sin  u,  [di/dt]00  and  [dQ/dt}00  incorrect. 

In  this  computation  the  device  was  for  the  first  time  applied  of  finding  the  root 
G  by  approximations  and  then  depressing  the  cubic  equation  and  solving  the  resulting 
quadratic  equation  directly.  When  a'  and  hence  g  is  large,  some  such  device  becomes 
necessary  as  the  solution  by  HILL'S  formulas  involves  a  great  amount  of  labor. 
Thus,  while  but  three  approximations  were  necessary  with  the  Earth  on  Mercury,  no 
less  than  eleven  were  required  in  some  cases  with  Mars  on  Mercury,  and  in  the  latter, 
as  well  as  in  the  present  case,  if  the  formulas  of  HILL'S  second  method  are  employed 
the  angle  6'  will  be  found  so  nearly  equal  to  90°  as  to  render  the  values  of  the  roots 
obtained  from  it  but  little  better  than  first  approximations.  Accordingly  all  the 
remaining  computations  have  been  effected  by  the  method  here  outlined,  a  method 
which,  since  the  approximation  to  the  value  of  G  is  always  very  rapid,  leads  so  quickly 
to  the  values  of  the  roots  that  special  devices  for  avoiding  the  solution  of  the  cubic 
seem  unnecessary. 

The  final  sums  are  here  practically  in  exact  agreement,  showing  that  the  effect 
of  all  terms  from  the  sixth  to  the  eleventh  orders  inclusive  is  inappreciable. 


82 


THE    SECULAH   VARIATIONS    OF   THE   ELEMENTS 


ACTION  OF  SATURN  ON  MERCURY. 


E 

A 

B  cos  t 

/.'  sin  . 

9 

h 

0° 

91.40055452 

7.9236831 

-0.7566872 

0.1637322 

90.704247 

22.5 

91.41316673 

8.0026121 

+0.6479546 

0.1200578 

90.704730 

45 

91.37113281 

7.5285556 

+  1.9837667 

1.1253368 

90.704973 

67.5 

91.28182268 

6.5736833 

+3.0473832 

2.6555561 

90.704844 

90 

91.15980423 

5.2833659 

+3.6768805 

3.8659866 

90.704445 

112.5 

91.02405557 

3.8540430 

+3.7764200 

4.0781377 

90.704075 

135 

90.89484135 

2.5033169 

+3.3308508 

3.1725737 

90.704038 

157.5 

90.79086206 

1.4368221 

+2.4080044 

1.6581198 

90.704442 

180 

90.72697664 

0.8169216 

+  1.1483775 

0.3771123 

90.705145 

202.5 

90.71250894 

0.7379931 

-0.2562644 

0.0187792 

90.705802 

225 

90.75006384 

1.2120495 

-1.5920765 

0.7248182 

90.706059 

247.5 

90.83489465 

2.1669211 

-2.6556933 

2.0167735 

90.705781 

270 

90.95505790 

3.4572383 

-3.2851902 

3.0861872 

90.705088 

292.5 

91.09266180 

4.8865615 

-3.3847297 

3.2760409 

90.704321 

315 

91.22635509 

6.2372881 

-2.9391612 

2.4702909 

90.703855 

337.5 

91.33481374 

7.3037834 

-2.0163140 

1.1625662 

90.703855 

2, 

728.48478638* 

34.9624190} 

+  1.5667604t 

14.9860379 

725.637848 

22 

728.48478617 

34.9624206 

+  1.5667608 

14.9860312 

725.637850 

*8ai  +  4ase*  +  8[a'J  -  2kaa'ee'  cos  A']  =  +  728.48478640. 
1 8(a'V  -  kaa'e  cos  K]  =  +  34.9624198. 
t  -  Kk'aa'  cos  »>'  •  e  sin  K'  =  +  1.5667610. 


OF   THE    ORBITS   OF   THE   FOUR    INNER   PLANETS. 


83 


E 


ACTION  OF  SATURN  ON  MERCURY. 
G  G' 


G" 


0 

0 

+0.410351 

90.704227 

0.4147226 

0.0043526 

0 

3 

53'  5078 

22.5 

+0.422480 

90.704715 

0.4256040 

0.0031100 

3 

56  31.53 

45 

+0.380202 

90.704836 

0.4105578 

0.0302188 

3 

59  47.96 

67.5 

+0.291021 

90.704520 

0.3703889 

0.0790439 

4 

2   4.89 

90 

+0.169403 

90.703974 

0.3081768 

0.1383038 

4 

1  12.34 

112.5 

+0.034023 

90.703579 

0.2300012 

0.1954823 

3 

55  22.98 

135 

-0.095154 

90.703652 

0.1455476 

0.2403156 

3 

44   5.05 

157.5 

-0.199537 

90.704241 

0.0683029 

0.2676388 

3 

29   2.13 

180 

-0.264126 

90.705099 

0.0149026 

0.2789825 

3 

15  29.12 

202 

.5 

-0.279250 

90.705800 

0.0007394 

0.2799874 

3 

11   3.15 

225 

-0.241953 

90.705971 

0.0294521 

0.2713169 

3 

17  46.29 

247.5 

-0.156844 

90.705536 

0.0901199 

0.2467190 

3 

29  20.24 

270 

-0.035987 

90.704713 

0.1675089 

0.2031209 

3 

39  39.24 

292.5 

+0.102383 

90.703922 

0.2482641 

0.1454820 

3 

46  28.97 

315 

+0.236543 

90.703554 

0.3215440 

0.0847000 

3 

50   7.92 

337.5 

+0.345001 

90.703713 

0.3789646 

0.0338216 

3 

52   2.72 

O 

/   // 

•v 

+  0.559279 

725.636026 

1.8124124 

1.2513111 

29 

41  58.70 

Zo 

+0.559277 

725.636026 

1.8123850 

1.2512850 

29 

41  56.61 

* 

ACTION  OF 

SATURN  ON  MERCURY. 

E 

log  A'o 

log  LO' 

log  #„ 

log  N        log  P 

logQ 

0° 

0.00150875 

0.27501245 

0. 

17835367    5.6285747    1.9882904 

3.8492800 

22.5 

0 

.00154356 

0. 

27505883 

0. 

17840585    5. 

6455608    2.0053300 

3.8663219 

45 

0 

.00158664 

0.27511623 

0.17847042    5.6919307    2.0514967 

3.912626(1 

67.5 

0.00161702 

0. 

27515671 

0. 

17851595    5. 

7568907    2.1160330 

3.9773995 

90 

0.00160533 

0.27514113 

0. 

17849842    5. 

8276384    2.1862035 

4.0478488 

112 

.5 

0.00152867 

0.27503898 

0. 

17838353    5. 

8929411    2.2508613 

4.1127652 

135 

0 

.00138529 

0.27484791 

0. 

17816860    5. 

9445922    2.3018923 

4.1639869 

157.5 

0 

.00120533 

0.27460806 

0. 

17789879    5. 

9773700    2.3341639 

4.1963617 

180 

0.00105402 

0.27440638 

0.17767193    5. 

9884810    2.3449566 

4.2071875 

202.5 

0.00100673 

0.27434335 

0. 

17760102    5.9770716    2.3334678 

4.1956990 

225 

0 

.00107884 

0. 

27443945 

0. 

17770914    5. 

9440470    2.3006205 

4.1628231 

247.5 

0 

.00120882 

0. 

27461270 

0.17790402    5. 

8922401    2.2492258 

4.1113306 

270 

0 

.00133101 

0. 

27477556 

0. 

17808722    5. 

8268942    2.1844672 

4.0463801 

292.5 

0 

.00141515 

0. 

27488768 

0. 

17821333    5. 

7562165    2.1144600 

3.9761077 

315 

0.00146116 

0. 

27494901 

0. 

17828232    5.6914235    2.0503133 

3.9116762 

337.5 

0.00148559 

0.27498156 

0. 

17831893    5. 

6452895    2.0046971 

3.8658214 

2! 

0 

.01101104 

1. 

19868812 

1. 

42524172    6.5435816    7.4082403 

2.3018085 

22      0.01101087 

1. 

19868787 

1. 

42524142    6. 

5435802    7.4082388 

2.3018068 

84 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  SATURN  ON  MERCURY. 


E 

logy 

Ji'              J2 

J, 

pt 

0° 

3.8492539 

90.058195      -0.59752575 

-  6.047237 

+  3.829727 

22.5 

3.8663033 

89.638288      -0.20995430 

-  7.910876 

-  3.279413 

45 

3.9124452 

89.669758      +0.36734212 

-  8.328272 

-10.040194 

67.5 

3.9769269 

90.090673      +0.71855783 

-  7.235732 

-15.423345 

90 

4.0470220 

90.582478      +0.67967125 

-  4.799462 

-18.609342 

112.5 

4.1115969 

90.879839      +0.35660262 

-  1.390299 

-19.113128 

135 

4.1625507 

90.892982      -0.04825504 

+  2.472681 

-16.858027 

157.5 

4.1947623 

90.670885      -0.36411899 

+  6.201244 

-12.187337 

180 

4.2055204 

90.332824      -0.49291773 

+  9.227617 

-  5.812141 

202.5 

4.1940258 

90.018885      -0.40939877 

+  11.090997 

+  1.296998 

225 

4.1612018 

89.853321      -0.15646464 

+  11.507770 

+  8.057780 

247.5 

4.1098562 

89.914463      +0.15541932 

+10.414614 

+  13.440932 

270 

4.0451660 

90.199819      +0.35977661 

+  7.978084 

+  16.626927 

292.5 

3.9752380 

90.576579      +0.29271946 

+  4.569179 

+  17.130715 

315 

3.9111697 

90.780941      -0.07365934 

+  0.706820 

+  14.875613 

337.5 

3.8656191 

90.580690      -0.50178023 

-  3.021123 

+  10.204923 

2, 

2.2943297 

722.370318*     +0.03796748 

+  12.718001 

-  7.929657 

22 

2.2943284 

722.370302      +0.03804694 

+  12.718004 

-  7.929655 

ACTION  OF  SATURN  ON  MERCURY. 

E 

Ft 

10000  X  fto     1000000  X  So    1000000  X  W, 

1000  X  ft'"' 

1000000  XS'»> 

0° 

-0.2217173 

0.20835581   -0.38501122    -  4.2759258 

0.0000000 

-1.2520309 

22.5 

+0.0670688 

0.21359630   -0.18752118    -  5.8140290 

+2.6067657 

-0.5980245 

45 

-0.2247552 

0.23766012   +0.18723103    -  6.8102258 

+5.0798256 

+0.5659595 

67.5 

-0.9885705 

0.27944181   +0.47990507    -  6.8742610 

+7.2389467 

+  1.3456245 

90 

-1.8298399 

0.33354940   +0.47167629    -  5.3763783 

+  8.6166490 

+  1.2184913 

112.5 

-2.2911663 

0.39056810   +0.12052582    •  1.8384740 

+8.6416600 

-0.2886458 

135 

-2.1148434 

0.43919690   -0.40799123   +  3.5527836 

+7.0044048 

-0.9201906 

157.5 

-1.3919042 

0.46950930   -0.83324243   +  9.6804438 

+3.9006063 

-1.8089208 

180 

-0.5106652 

0.47592790   -0.91983080   +14.8005740 

0.0000000 

-1.9709759 

202.5 

+0.0653940 

0.45862525   -0.61203702   +17.3393050 

-3.8101033 

-1.3286969 

225 

+0.0610980 

0.42282850   -0.06578321   +16.6811060 

-6.7433594 

-0.1483686 

247.5 

-0.4587079 

0.37651300   +0.43874423   +13.4040390 

-8.3306788 

+  1.0507430 

270 

-1.1366346 

0.32772690   +0.65346841   +  8.8350920 

-8.4662365 

+  1.6881183 

292.5 

-1.5401525 

0.28276850   +0.49946088   +  4.2959022 

-7.3251220 

+  1.4004581 

315 

-1.4203565 

0.24587472   +0.10699355   +  0.5601261 

-5.2554073 

+0.3234187 

337.5 

-0.8596732 

0.22009701   -0.26508243    -  2.2258014 

-2.6861019 

-0.8453756 

2, 

-7.3977141 

2.69112025   -0.35924718   +27.9671518 

+0.2358762 

-0.4955782 

22 

-7.3977118 

2.69111927   -0.3592470?   +27.9671246 

+0.2365727 

-0.4955464 

sin 

<f-  j^li'"  +  COS  <f-l 

Vc)  =  +  0.000000000034. 

*2j 

(Ji'  -  G")  =  721. 

119007.  22(J/  -  G")  =  721.119017. 

OF   THE    ORBITS    OF   THE    FOUR   INNER   PLANETS. 


85 


E 


1000  X  [ft)  sin  v 

+  (COS  V  +  COS  E)S0] 


1000X 


ACTION  OF  SATURN  ON  MERCURY. 

[ 
"COS" 


o 


1000000  XWt  cos  u      1000000  X  Wo  sin  u 


1000  X  -2-  Bo 
a 


0° 

-0.000770023 

-0.0208355810 

-  3.7553310 

-  2.0447571 

-0.033103367 

22.5 

+0.009535644 

-0.0190997850 

-  3.2422806 

-  4.8260289 

-0.034604548 

45 

+0.019486084 

-0.0136593860 

-  0.8728046 

-  6.7540641 

-0.040621607 

67.5 

+0.027699067 

-0.0044469025 

+  2.0656152 

-  6.5565758 

-0.051490988 

90 

+0.032545329 

+0.0078015052 

+  3.4868920 

•  4.0923123 

-0.066709877 

112.5 

+0.032625198 

+0.0215155020 

+  1.6174933 

-  0.8739008 

-0.084259712 

135 

+0.027148276 

+  0.0344565700 

-  3.5128323 

+  0.5313050 

-0.100609845 

157.5 

+0.016337313 

+0.0439769600 

-  9.5267356 

-   1.7182294 

-0.111738865 

180 

+0.001839662 

+0.0475927900 

-12.9986018 

-  7.0776661 

-0.114756176 

202.5 

-0.013287669 

+0.0439638930 

-11.8447875 

-12.6630324 

-0.109148595 

225 

-0.025446823 

+0.0337808020 

-  6.8547683 

-15.2076154 

-0.096860250 

247.5 

-0.031966152 

+0.0197522140 

•   1.0475614 

-13.3630400 

-0.081227519 

270 

-0.032206863 

+0.0054309792 

+  2.5393222 

-  8.4623118 

-0.065545379 

292.5 

-0.027462505 

-0.0063965152 

+  2.7411608 

-  3.3076901 

-0.052103952 

315 

-0.019770536 

-0.0145922640 

+  0.5055592 

-  0.2411456 

-0.042025673 

337.5 

-0.010655672 

-'0.0192899750 

-  2.2254410 

-  0.0400332 

-0.035657730 

2i 

+0.002825107 

+0.0799753913 

-21.4625646 

-43.3484220 

-0.560232174 

22 

+0.002825224 

+0.0799754154 

-21.4625365 

-43.3485702 

-0.560231909 

DIFFERENTIAL  COEFFICIENTS. 

// 

log  coeff. 

[de/dJfe     =  +     1 

.8596825  m'    p 

0.2694389 

[dx/dt]M    =  +256 

.04618      m'     p 

2.4083183 

[di/dt]w     =        14 

.751452    m'    n 

1.1688348 

[da/dflM    =  -244 

.39983      m'    n 

2.3881009 

[dir/dt]m    =  +254 

.22335      m'     p 

2.4052154 

[dL/dfloo   =  -373 

.17967      m'    n 

2.5719180 

FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  m'. 

u 

[de/df]«)    = 

+0.00053109524 

[dx/d*]M    = 

+0.073122627 

[di/dt]M    = 

-0.0042127757 

[dtt/dt]M    = 

-0.069796619 

[dir/dt]o0    = 

+0.072602050 

[dL/dtlw   = 

-0.10657405 

86 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


[de/dt], 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss, 

oo 


sin  i  [dtt/dt]m 


+o'.00053 

+0.00053 

+0.0005311 

+0.01494 

+0.01503 

+0.0149273 

-0.00421 

-0.00421 

-0.0042128 

-0.00853 

-0.00850 

-0.0085087 

-0.1070 

-0.106574 

NOTES. 

The  considerable  disagreement  of  the  first  sums  is  caused  by  the  rather  large 
value  of  e'.  The  very  exact  agreement  toward  the  close  of  the  computation  shows, 
however,  that  all  terms  above  the  15th  order  are  wholly  inappreciable,  the  total 
effect  of  all  terms  from  the  8th  to  the  15th  orders  inclusive  occurring  with  [de/dt]0o 
and  amounting  to  but  1 /30000th  of  the  value  of  this  coefficient. 


ACTION  OF  URANUS  ON  MERCURY. 


E 

A 

B  cos  e 

B  sin  . 

a 

h 

0° 

368.36907643 

16.94656315 

-5.8755608 

27.995613 

367.49553 

45 

368.87526360 

22.1817430 

-3.9726882 

12.798540 

367.49652 

90 

369.14240318 

24.5977597 

+  1.1481399 

1.069008 

367.49606 

135 

369.01848670 

22.7793354 

+6.4872119 

34.127748 

367.49535 

180 

368.57162431 

17.7916836 

+8.9169755 

64.480235 

367.49625 

225 

368.05910248 

12.5565017 

+7.0141016 

39.896574 

367.49742 

270 

367.78562896 

10.1404905 

+  1.8932751 

2.906824 

367.49671 

315 

367.91587927 

11.9589113 

-3.4457976 

9.628773 

367.49535 

2, 

1473.86873228* 

69.4764969t 

+6.0828297J 

96.451680 

1469.98455 

22 

1473.86873205 

69.4764914 

+6.0828277 

96.451635 

1469.98464 

*  4a2  +  2aV  +  4[a'2  -  Zkoa'ee'  cos  K]  =  +  1473.86873246. 
t  4[a'V  -  kaa'e  cos  A']  =  +  69.4764933. 
t  -  4k'aa'  cos  J  •  e  sin  A"  =  +  6.0828300. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


87 


ACTION  OF  URANUS  ON  MERCURY. 


E 

I 

G             G'           G" 

8 

O      /       U 

0° 

+0.06261 

367.49532       0.30919       0.24638 

2  13  39.23 

45 

+0.56779 

367.49643       0.62373       0.05584 

2  27  51.84 

90 

+0.83539 

367.49605       0.83888       0.00347 

2  44  38.85 

135 

+0.71220 

367.49510       0.82501       0.11257 

2  53  41.24 

180 

+0.26443 

367.49577       0.57178       0.30687 

2  48   5.47 

225 

-0.24927 

367.49712       0.22774       0.47670 

2  30  27.70 

270 

-0.52202 

367.49669       0.01474       0.53674 

2  13   6.46 

315 

-0.39041 

367.49528       0.05839       0.44873 

2   7  39.31 

2, 

+0.64041 

1469.98383       1.73458       1.09346 

9  59  30.01 

22 

+0.64031 

1469.98393       1.73486       1.09383 

9  59  40.09 

E 

log  jr. 

log  L0'         log  No         log  N         log  P 

logQ 

0° 

0.00049251 

0.27365789    0.17682994    4.7157168    9.8582891 

2.3270039 

45 

0.00060285 

0.27380499    0.17699541    4.7796309    9.9227977 

2.3913072 

90 

0.00074754 

0.27399787    0.17721240    4.9163276    0.0598121 

2.5282833 

135 

0.00083193 

0.27411036    0.17733893    5.0341227    0.1774639 

2.6460770 

180 

0.00077916 

0.27404002     0.17725980     5.0782321     0.2210425 

2.6898771 

225 

0.00062422 

0.27383348    0.17702747    5.0332669    0.1754670 

2.6444777 

270 

0.00048849 

0.27365254    0.17682391    4.9151227    0.0570006 

2.5260593 

315 

0.00044928 

0.27360025     0.17676511     4.7787835     9.9208203 

2.3897669 

Zj 

0.00250770 

1.09534833     0.70812605     9.6253991     0.1961442 

0.0712235 

22 

0.00250828 

1.09534908     0.70812692     9.6258039     0.1965489 

0.0716217 

E 

logF 

Ji'            Ji             J3 

Fi 

0° 

2.3266400 

366.509888      -2.2679092      -1.6777615 

+  100.81429 

45 

2.3912247 

363.119586      -0.5138777      -3.4254958 

+  68.16435 

90 

2.5282782 

365.160838      +2.2768225      -2.9100648 

•  19.70006 

135 

2.6459108 

367.567473      +0.7238013      -0.4333073 

-111.30915 

180 

2.6894241 

366.570373      -1.5747120      +2.5538323 

-152.99965 

225 

2.6437740 

364.094438      -1.1669936      +4.3014333 

-120.34967 

270 

2.5252669 

364.090394      +1.5139175      +3.7858697 

-  32.48527 

315 

2.3891044 

367.299619      +1.4067899      +1.3092445 

+  59.12384 

Si 

0.0696092 

1462.331493*     -0.0518812      +1.7518757 

-104.37069 

2, 

0.0700139 

1462.082117      -0.4497199      +1.7518747 

-104.37063 

*Si(J,'  -  G")  =  1461.238038. 

Z2(./i'  -  G")  =  1460.987287. 

THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  UKANUS  ON  MERCURY. 

E 

F3                 1000000  X  Ro     1000000000  X  <S0        1000000  X  TFo          100000  X  ft"" 

1000000  X  S'"> 

0° 

-  7.546840         2.572319         -  40.838728        -0.3564792            0.00000000 

-0.13280483 

45 

-  0.473969         2.902866               6.943520        -0.8432739         +0.62046843 

-0.02098878 

90 

-   1.504590         4.048391        +  74.582644        -0.9823294         +1.04582909 

+0.19267091 

135 

-14.059822         5.410853        +  15.278801        -0.1938530         +0.86293520 

+0.03446006 

180 

-17.382080         5.928252        -102.476485        +1.2462639            0.00000000 

-0.21958237 

225 

-  5.500356         5.225873         -  69.412255        +1.8932075         -0.83343288 

-0.15655363 

270 

+  1.222261         3.978026        +  47.038051        +1.2690546         -1.02765154 

+0.12151438 

315 

-  5.177088         2.987389        +  39.388463        +0.3202880         -0.63853471 

+0.11906293 

£, 

-25.211249        16.526988         -  21.694518        +1.1765099         +0.01817755 

-0.03820191 

Ii 

-25.211235        16.526981          -  21.688511        +1.1763686         +0.01143604 

-0.02401942 

JS 

lOOOOOOXlffi  sin«4-    1000000><     -#ocosv+ 
OM.  +  WJW   /'                 \.          -,    lOOOOOOXTF.coau          1000000  XTF0  sin  „ 
(  -sec2v>  +  ll  sin  vS0\ 

1000000  X-  2-  ft 
a 

0° 

-0.0816775            -2.5723195            -0.31307783            -0.17046914 

4.0868774 

45 

+2.3415248             -1.7140875             -0.10807474             -0.83631981 

-  4.9616693 

90 

+3.9465628            +0.9815683             +0.63709735             -0.74771500 

-   8.0967815 

135 

+3.2460626            +4.3319665            +0.19167313             -0.02898997 

-12.3950199 

180 

+0.2049530            +5.9282527            -1.09453081            -0.59596603 

-  14.2942.r)«M> 

225 

-3.0528880            +4.2563802            -0.77797582             -1.72597460 

-11.9712672 

270 

-3.9027069            +0.7238029            +0.36474303             -1.21550896 

-   7.9560519 

315 

-2.3679844             -1.8134015            +0.28908589             -0.13789041 

-  5.1061400 

2! 

+0.1671314            +5.0613044            -0.40576826             -2.72965913 

-34.4339707 

22 

+0.1667150            +5.0608577            -0.40529154            -2.72917479 

-34.4340964 

sin  p  •  1^,0  +  cos  v  •  Bo(c>  =  -  0.00000000000024. 

DIFFERENTIAL  COEFFICIENTS. 

„                          log  coeff. 

[de/dt]m    =  +  0.21975650  TO'     p  9.3419417 

[dxldt]m    =  +32.406731      TO'     p  1.5106352 

[dildt]M    =--  0.55745051  TO'    »  9.7462063 

[rfn/<ft]M    =  -30.777028      TO'    n  1.4882267 

[dw/dt]^    =  +32.177180      TO'     p  1.5075480 

[dLldt]M   =  -45.859693      TO'    n  1.6614312 

OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  89 

FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  m'. 


=  +0.0000096384435 
[dx/dt]m    =  +0.0014213479 
[dtYdflw     =  -0.000024449584 
[dQ/dt]m    =  -0.0013498699 
[dT/dfloo    =  +0.0014112801 
[dL/dt]m  =  -0.0020113907 

COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

//  //  // 

{de/dt]m  +0.00000  +0.00001  +0.0000096 

e[dw/dt]w  +0.00029  +0.00029  +0.0002902 

[dt'/dfloo  -0.00001  -0.00002  -0.0000244 

sin  i  [dtt/dt]w  -0.00016  -0.00016  -0.0001646 

NOTES. 

In  the  results  of  this  computation,  published  in  A.  J.,  No.  398,  the  residual  from 
the  test  equation  which  arises  from  the  constancy  of  the  major  axis  was  stated  very 
much  too  large.  Its  true  value  is  as  here  given. 

A  comparison  of  the  above  figures  with  the  corresponding  tabulation  for  Saturn 
on  Mercury  and  Uranus  on  Venus  shows  'that  a  division  into  eight  parts  is  fully 
sufficient.  The  effect  of  all  terms  of  the  4th  and  higher  orders  may,  however,  in 
some  cases  amount  to  l/1000th  of  the  whole. 

ACTION  OF  NEPTUNE  ON  MERCURY. 


E 

A 

B  COS  e 

Bsin  e 

g 

h 

0° 

904.45979027 

+  15.1625927 

-  5.380877 

1.889897 

904.17356 

45 

904.50658592 

+  17.0408109 

+  3.150684 

0.647955 

904.17446 

90 

904.46648308 

+  12.3030460 

+  10.652425 

7.406776 

904.17363 

135 

904.36745259 

+  3.7246157 

+  12.729927 

10.577523 

904.17288 

180 

904.26302599 

-  3.6693551 

+  8.166225 

4.352861 

904.17435 

225 

904.20989585 

-  5.5475736 

-  0.365336 

0.008712 

904.17570 

270 

904.24366422 

-  0.8098086 

-  7.867076 

4.039717 

904.17446 

315 

904.34902898 

+  7.7686225 

-  9.944584 

6.455136 

904.17292 

Si 

3617.43296356* 

+22.9864685f 

+  5.5706981 

17.689251 

3616.69599 

22 

3617.43296334 

+22.9864755 

+  5.570691 

17.689326 

3616.69596 

*  4o2  +  2aV  +  4[a'2  -  2kaa'ce'  cos  K]  =  +  3617.4329635, 
t  4[a'V  -  kaa'e  cos  A']  =  +  22.986469. 
J  —  4fc'oo'  cos  <f   •  e  sin  A''  =  +  5.570695, 


90 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  NEPTUNE  ON  MERCURY. 


E 

I 

G 

G' 

G" 

e 

0      1       II 

0° 

0.22096 

904.17356 

0.2300460 

0.0090860 

0  55  54.56 

45 

0.26686 

904.17446 

0.2695190 

0.0026589 

0  59  38.88 

90 

0.22759 

904.17362 

0.2591945 

0.0316047 

1   1  39.23 

135 

0.12931 

904.17286 

0.1906740 

0.0613537 

0  57  23.73 

180 

0.02341 

904.17434 

0.0820755 

0.0586556 

0  42  53.31 

225 

0.03107 

904.17570 

0.0003070 

0.0313770 

0  20  20.99 

270 

0.00394 

904.17445 

0.0688415 

0.0649018 

0  41  48.59 

315 

0.11084 

904.17291 

0.1564755 

0.0456255 

0  51  23.82 

s, 

0.57589 

3616.69597 

0.6401575 

0.1642481 

3  22  15.69 

22 

0.57594 

3616.69593 

0.6169755 

0.1410151 

3   8  47.42 

E 

log  A'o 

log  /,„' 

logtfo 

log  N        log  P 

logQ 

0° 

0.00008616 

0.27311614 

0.17622049     4 

.1292404     8.4898441 

1.3492047 

45 

0.00009807 

0.27313202 

0.17623836    4 

.1927246     8.5533496 

1.4127095 

90 

0.00010478 

0.27314097 

0.17624841     4 

.3291694    8.6897764 

1.5491508 

135 

0.00009080 

0.27312233 

0.17622746    4.4470372    8.8075977 

1.6669838 

180 

0.00005070 

0.27306887 

0.17616731     4 

.4915054     8.8520137 

1.7113924 

225 

0.00001141 

0.27301649 

0.17610837    4 

.4469773     8.8074580 

1.6668178 

270 

0.00004818 

0.27306552 

0.17616353    4 

.3290882     8.6895869 

1.5489683 

315 

0.00007281 

0.27309835 

0.17620048     4, 

.1926694    8.5532210 

1.4125965 

2, 

0.00028982 

1.09239150 

0.70479974    7.2790034    4.7212211 

6.1587162 

22 

0.00027309 

1.09236919 

0.70477467    7 

.2794084    4.7216261 

6.1591075 

E 

logF 

J,' 

J. 

J3 

F, 

0° 

1.3491992 

897.83542 

-6.8010114 

-  59.717292 

+41.026915 

45 

1.4127079 

890.85490 

+  1.7283795 

-  93.077013 

-24.022644 

90 

1.5491318 

899.76750 

+6.4446186 

-  62.851349 

-81.220264 

135 

1.6669469 

904.08417 

-1.3097917 

+  13.254268 

-97.060333 

180 

1.7113571 

897.88500 

-6.6871696 

+  90.657875 

-62.264029 

225 

1.6667989 

891.04449 

-2.2340656 

+  124.017084 

+  2.785531 

270 

1.5489293 

894.36354 

+5.9480179 

+  93.790933 

+59.983153 

315 

1.4125691 

903.73833 

+2.4206062 

+  17.685807 

+75.823263 

2! 

6.1586174 

3589.85146* 

-1.0955445 

+  61.880167 

-42.474225 

22 

6.1590228 

3589.72189 

+0.6051284 

+  61.880146 

-42.474183 

-G")  =  3589.68721. 
-  G")  =  3589.58087. 


OF    THE    ORBITS    OF   THE    FOUR    INNER    PLANETS. 


91 


ACTION  OF  NEPTUNE  ON  MERCURY. 

E 

F  j                  1000000  X  Ko     1000000000  X  So     1000000000  X  W0     1000000  X  fi"° 

1000000  X  <S(n) 

0° 

-  2.928627 

0.6591808         -15.070783         -133.45316            0.000000 

-0.04900917 

45 

-    0.320722 

0.7449111         +  4.384511         -240.74232         +1.592198 

+0.01325344 

90 

-  8.201274 

1.0513250         +22.423107         -222.59983         +2.715911 

+0.05792607 

135 

-13.630154 

1.3989891           -  6.706655         +  61.47306         +2.231138 

-0.01512625 

180 

-  6.745208 

1.5178690         -34.846090         +466.35639            0.000000 

-0.07466679 

225 

+  0.065738 

1.3382857         -10.354891         +575.81202         -2.134326 

-0.02335460 

270 

-  3.847073 

1.0317860         +21.345904         +331.94457         -2.665434 

+0.05514330 

315 

-  7.846125 

0.7779772        +  6.529864        +  45.70113         -1.662874 

+0.01973839 

r, 

-21.722182 

4.2601608          -  6.147862        +442.24897        +0.050477 

-0.01060659 

22 

-21.731263 

4.2601631           -  6.147171         +442.24389         +0.026136 

-0.00548902 

E 

1000000000  Xtffo  sm  v  lwooooooo><L-tf<>c°s<'        1000000000                    1000000000 

1000000000 

7* 

+  (cosw+cos£)<S 

o]         fr      2          \   .      „"!             XWoCOsw                     XWosmu 

x-afa 

0° 

30.1416 

-  659.1808              -117.20523                -  63.81760 

-  1047.3000 

45 

+  608.8428 

-  430.4075               -  30.85375             -238.75703 

-1273.2255 

90 

+  1024.2533 

+  261.0141              +144.36884              -169.43527 

-2102.6500 

135 

+  855.3050 

+  1105.9013                -  60.78178              +     9.19306 

-3204.7615 

180 

+     69.6922 

+  1517.8690              -409.57745              -223.01267 

-3659.9008 

225 

-  792.9698 

+  1080.1647              -236.61842              -524.94878 

-3065.7028 

270 

-1014.1308 

+   169.4383              +  95.40526              -317.93876 

-2063.5719 

315 

-  621.4949 

-  466.5348              +  41.24897                -   19.67525 

-1329.7429 

Si 

+     49.6731 

+  1289.1406             -287.00858             -774.20430 

-8873.4227 

S2 

+     49.6831 

+  1289.1237             -287.00498             -774.18800 

-8873.4327 

sin  <f  •  \A  iw  +  cos  if  •  L 

t0(c)  =  +  0.000000000000013. 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[deldt]<n    =  +  o'.065401848  TO'     p  8.8155900 

[dx/dt]m    =  +  8.2544736      TO'     p  0.9166894 

[di/dt]w     =  -  0.39452600    TO'     n  9.5960756 

[drfdt] 


w 


=  -  8.7298700      TO' 
=  +  8.1893623      TO' 


n  0.9410078 
p  0.9132501 


[dL/d*]oo   =  -11.826130       TO'    n  1.0728427 


92  THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  m'. 

[de/dt}00  =  +0.00000331989 
[dx/dtlw,        +0.00041900885 

[di/dt]oo  =  -0.00002002670 

[dV/dt]w  =  -0.00044314061 

[dw!dt]m  =  +0.00041570371 

[dL/dt]w  =  -0.00060031125 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


[de/dt]w    +o'.()0000 

+0.00000 

+o'.0000033 

c[d7r/d<]oo    +0.00009 

+0.00009 

+0.0000855 

[di/dt]m    -0.00001 

-0.00002 

-0.0000200 

sinz   [dtt/dt]m    -0.00005 

-0.00005 

-0.0000508 

NOTES. 

In  the  final  results  of  this  computation,  published  in  A.  J.,  No.  398,  the  value  of 
the  residual  arising  from  the  equation  [da/dt]00  =  0  is  greatly  overstated.  Its  true 
value  is  that  given  above. 

The  very  large  disagreement  in  G',  G",  6,  etc.,  arises  from  the  large  values  of  e' 
and  I  but  the  gradual  lessening  of  the  discrepancies  as  the  end  of  the  computation  is 
approached  shows  that  terms  of  the  8th  and  higher  orders  are  wholly  inappreciable. 
The  greatest  effect  produced  by  all  terms  of  the  4th  and  higher  orders  here  occurs 
with  [de/dt]00  and  amounts  to  but  1 /10000th  of  the  value  of  this  coefficient. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


93 


VENUS. 
ACTION  OF  MEKCURY  ON  VENUS. 


E 

A 

B  cos  « 

B  gin  e 

1000  x  g 

0° 

0.73249627 

+0.19271542 

+0.22036223 

0.30759837 

15 

0.70628935 

+0.12839518 

+0.25427819 

0.40956972 

30 

0.67778527 

+0.05734848 

+0.27076206 

0.46439247 

45 

0.64892716 

-0.01558282 

+0.26869056 

0.45731389 

60 

0.62168257 

-0.08542863 

+0.24820463 

0.39023784 

75 

0.59791027 

-0.14742905 

+0.21070058 

0.28121662 

90 

0.57923027 

-0.19735886 

+0.15873412 

0.15960636 

105 

0.56691636 

-0.23181556 

+0.09584678 

0.05819218 

120 

0.56180731 

-0.24845074 

+0.02632410 

0.00438952 

135 

0.56425026 

-0.24613080 

-0.04509594 

0.01288203 

150 

0.57407787 

-0.22501408 

-0.11354628 

0.08166866 

165 

0.59061890 

-0.18653936 

-0.17436209 

0.19258116 

180 

0.61274521 

-0.13332873 

-0.22339892 

0.31613445 

195 

0.63894853 

-0.06900848 

-0.25731488 

0.41941058 

210 

0.66744400 

+0.00203821 

-0.27379880 

0.47486769 

225 

0.69628968 

+0.07496950 

-0.27172719 

0.46770903 

240 

0.72352217 

+0.14481528 

-0.25124139 

0.39984532 

255 

0.74728586 

+0.20681571 

-0.21373734 

0.28938127 

270 

0.76596211 

+0.25674559 

-0.16177082 

0.16577156 

285 

0.77827946 

+0.29120222 

-0.09888350 

0.06193800 

300 

0.78339732 

+0.30783739 

-0.02936080 

0.00546066 

315 

0.78096661 

+0.30551754 

+0.04205926 

0.01120554 

330 

0.77115125 

+0.28440072 

+0.11050957 

0.07735875 

345 

0.75461912 

+0.24592608 

+0.17132542 

0.18593159 

2i 

8.07130162* 

0.356320051 

-0.018220301 

2.84733165 

St 

8.07130156 

0.35632016 

-0.01822015 

2.84733161 

*  12a2  +  6aV  +  12[a'2  - 

-  2kaa'ee'  cos  K]  =  8.07130158. 

1  12[a'V  -  kaa'e  cos  K] 

=  +  0.35632010. 

t  —  12fc'aa'  cos  <p'  •  e  sin 

K'  =  -  0.01822024. 

0.58840054 
0.56402284 
0.53673295 
0.50812387 
0.48000449 
0.45436878 
0.43333042 
0.41895978 
0.41292353 
0.41598961 
0.42773114 
0.44670373 
0.47092914 
0.49830819 
0.52683118 
0.55465379 
0.58013161 
0.60184004 
0.61860745 
0.62954599 
0.63407808 
0.63196045 
0.62329194 
0.60851491 

6.33299247 
6.33299197 


94 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


E 


ACTION  OF  MERCUKY  ON  VENUS. 
G  G' 


G" 


0 

0.13776127 

0.58723527 

0.14259981 

0.00367327 

29 

50 

12.97 

15 

0.13593205 

0.56231468 

0.14274284 

0.00510263 

30 

41 

37.56 

30 

0.13471786 

0.53456025 

0.14296704 

0.00607649 

31 

40 

18.70 

45 

0.13446883 

0.50568767 

0.14321940 

0.00631437 

32 

42 

44.77 

60 

0.13534362 

0.47761736 

0.14342737 

0.00569662 

33 

44 

34.65 

75 

0.13720703 

0.45239646 

0.14351084 

0.00433149 

34 

40 

36.18 

90 

0.13956539 

0.43206750 

0.14340426 

0.00257595 

35 

25 

4.49 

105 

0.14162212 

0.41845750 

0.14309623 

0.00097183 

35 

52 

44.73 

120 

0.14254931 

0.41288415 

0.14266320 

0.00007451 

36 

0 

33.62 

135 

0.14192619 

0.41587654 

0.14225701 

0.00021775 

35 

48 

50.91 

150 

0.14001227 

0.42706500 

0.14202490 

0.00134648 

35 

20 

41.47 

165 

0.13758072 

0.44529837 

0.14203102 

0.00304495 

34 

40 

10.49 

180 

0.13548161 

0.46890720 

0.14224327 

0.00473972 

33 

51 

11.13 

195 

0.13430588 

0.49597000 

0.14257523 

0.00593116 

32 

57 

11.97 

210 

0.13427836 

0.52451120 

0.14293246 

0.00633413 

32 

1 

25.39 

225 

0.13530143 

0.55262582 

0.14323803 

0.00590863 

31 

6 

52.56 

240 

0.13705610 

0.57856627 

0.14343948 

0.00481804 

30 

16 

21.83 

255 

0.13911137 

0.60079677 

0.14351100 

0.00335627 

29 

32 

27.98 

270 

0.14102020 

0.61804514 

0.14345226 

0.00186975 

28 

57 

29.65 

285 

0.14239902 

0.62934391 

0.14328793 

0.00068684 

28 

33 

26.40 

300 

0.14298478 

0.63406059 

0.14306246 

0.00006020 

28 

21 

52.79 

315 

0.14267170 

0.63192420 

0.14283210 

0.00012415 

28 

23 

50.37 

330 

0.14152485 

0.62303406 

0.14265313 

0.00087040 

28 

39 

39.46 

345 

0.13976975 

0.60786141 

0.14256873 

0.00214548 

29 

8 

52.62 

Si 

1.66229562 

6.31855399 

1.71486964 

0.03813556 

384 

9 

26.15 

S2 

1.66229608 

6.31855323 

1.71487036 

0.03813555 

384 

9 

26.54 

OF   THE    ORBITS   OF   THE    FOUR   INNER    PLANETS. 


95 


ACTION  OF  MEKCURY 

ON  VENUS. 

E 

log  K, 

log  Lo' 

logtfo 

logtf 

logP 

logQ 

0° 

0.09428087 

0.39679023 

0.31472571 

0.0090498 

0.8627996 

0.5522553 

15 

0.10017341 

0.40440450 

0.32320948 

0.0415726 

0.9381720 

0.6108796 

30 

0.10716555 

0.41342180 

0.33324961 

0.0806578 

1.0282684 

0.6810018 

45 

0.11492344 

0.42340404 

0.34435531 

0.1248163 

1.1296769 

0.7598999 

60 

0.12293987 

0.43369427 

0.35579389 

0.1716324 

1.2368681 

0.8431970 

75 

0.13049856 

0.44337414 

0.36654485 

0.2174860 

1.3415449 

0.9243732 

90 

0.13670353 

0.45130400 

0.37534554 

0.2575171 

1.4325547 

0.9947294 

105 

0.14065788 

0.45634998 

0.38094249 

0.2862203 

1.4972527 

1.0445040 

120 

0.14178785 

0.45779080 

0.38254019 

0.2989083 

1.5248859 

1.0655419 

135 

0.14009658 

0.45563407 

0.38014861 

0.2935151 

1.5107656 

1.0544719 

150 

0.13608370 

0.45051253 

0.37446742 

0.2714379 

1.4582282 

1.0140442 

165 

0.13043971 

0.44329885 

0.36646127 

0.2367595 

1.3768369 

0.9516100 

180 

0.12381666 

0.43481823 

0.35704266 

0.1945715 

1.2784801 

0.8761594 

195 

0.11676724 

0.42577304 

0.34698954 

0.1495758 

1.1741125 

0.7959472 

210 

0.10975153 

0.41675185 

0.33695547 

0.1054452 

1.0722608 

0.7174326 

225 

0.10314673 

0.40824135 

0.32748247 

0.0647779 

0.9789193 

0.6452104 

240 

0.09725105 

0.40063000 

0.31900464 

0.0292994 

0.8980200 

0.5823493 

255 

0.09229597 

0.39422222 

0.31186325 

0.0001263 

0.8320547 

0.5308426 

270 

0.08845847 

0.38925289 

0.30632243 

9.9779738 

0.7825625 

0.4919641 

285 

0.08587486 

0.38590395 

0.30258708 

9.9633062 

0.7504867 

0.4665316 

300 

0.08464926 

0.38431433 

0.30081370 

9.9564277 

0.7363980 

0.4550694 

315 

0.08485629 

0.38458290 

0.30111334 

9.9575313 

0.7406137 

0.4578944 

330 

0.08653835 

0.38676424 

0.30354673 

9.9667119 

0.7632399 

0.475140o 

345 

0.08969689 

0.39085723 

0.30811150 

9.9839477 

0.8041353 

0.5067244 

Zi 

1.32942669 

5.01604517 

4.05980799 

1.3196325 

13.0745661 

8.7488847 

Z2 

1.32942756 

5.01604627 

4.05980919 

1.3196348 

13.0745711 

8.7488890 

96 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  MERCURY  ON  VENUS. 


E 

log  V 

/»' 

Ji. 

J3 

1000  X  F, 

0° 

0.5489818 

0.14641488 

+0.046569366 

+0.007078444 

-6.6249758 

15 

0.6061455 

0.14799570 

+0.053805593 

+0.005998321 

-7.6446250 

30 

0.6750909 

0.14917691 

+0.057280924 

+0.004274648 

-8.1401963 

45 

0.7534266 

0.14962155 

+0.056746961 

+  0.002024896 

-8.0779189 

60 

0.8370288 

0.14915409 

+0.052252409 

-0.000597633 

-7.4620305 

75 

0.9194294 

0.14784239 

+0.044134510 

-0.003414224 

-6.3345072 

90 

0.9916527 

0.14602955 

+0.032988273 

-0.006232935 

-4.7721868 

105 

1.0433053 

0.14427332 

+0.019614339 

-0.008861684 

-2.8815397 

120 

1.0654487 

0.14317010 

+0.004953440 

-0.011121317 

-0.7914084 

135 

1.0542014 

0.14310862 

-0.009985226 

-0.012857840 

+  1.3557653 

150 

1.0124149 

0.14408811 

-0.024195183 

-0.013952904 

+3.4136578 

165 

0.9480753 

0.14573205 

-0.036737898 

-0.014331875 

+  5.2420265 

180 

0.8709298 

0.14748132 

-0.046798849 

-0.013968920 

+6.7162708 

195 

0.7897501 

0.14882171 

-0.053732494 

-0.012888777 

+7.7359211 

210 

0.7111602 

0.14942892 

-0.057095855 

-0.011165046 

+  8.2314924 

225 

0.6396416 

0.14920890 

-0.056671026 

-0.008915214 

+8.1692132 

240 

0.5779992 

0.14827058 

-0.052477157 

-0.006292610 

+7.5533276 

255 

0.5279163 

0.14686716 

-0.044770588 

-0.003475962 

+  6.4258044 

270 

0.4903757 

0.14532896 

-0.034036602 

-0.000657227 

+4.8634820 

285 

0.4659576 

0.14399775 

-0.020965237 

+0.001971497 

+2.9728356 

300 

0.4550194 

0.14316554 

-0.006416142 

+0.004231074 

+0.8827039 

315 

0.4577910 

0.14302191 

+0.008631431 

+0.005967519 

-1.2644706 

330 

0.4744060 

0.14361483 

+0.023141851 

+0.007062502 

-3.3223626 

345 

0.5048724 

0.14483260 

+0.036096092 

+0.007441419 

-5.1507310 

2, 

8.7105081 

1.75532379* 

-0.003833505 

-0.041341924 

+0.5477741 

2* 

8.7105125 

1.75532376 

-0.003833543 

-0.041341924 

+0.5477737 

*  2,(J,'  -  G")  =  1 

.71718823. 

S2(j2'  _  G")  =  1 

.71718811. 

OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


97 


ACTION  OF  MERCURY  ON  VENUS. 

E 

1000  X  F3 

R0 

So 

W0 

BW 

£<"> 

0° 

+0.12855762 

-1.0113711 

+0.11654330 

+0.02599380 

0.0000000 

+0.16223015 

15 

+0.28853773 

-1.0855630 

+0.15095431. 

+0.02672256 

-0.3910151 

+0.21008155 

30 

+0.43551240 

-1.1839703 

+0.18420453 

+0.02487781 

-0.8232930 

+0.25617923 

45 

+0.52994293 

-1.3088599 

+0.21275230 

+0.01862044 

-1.2857216 

+0.29555816 

60 

+0.54641763 

-1.4594255 

+0.23029016 

+0.00532100 

-1.7533282 

+0.31946706 

75 

+0.48046967 

-1.6276742 

+0.22753508 

-0.01781201 

-2.1774255 

+0.31512321 

90 

+0.34977540 

-1.7935752 

+0.19439994 

-0.05167293 

-2.4796011 

+0.26875615 

105 

+0.18941983 

-1.9243133 

+0.12616235 

-0.09195653 

-2.5651527 

+0.17410988 

120 

+0.04248891 

-1.9847155 

+0.03108865 

-0.12787906 

-2.3681413 

+0.04283321 

135 

-0.05148113 

-1.9570464 

-0.06917608 

-0.14733858 

-1.9039342 

-0.09517474 

150 

-0.06711060 

-1.8525243 

-0.15091802 

-0.14550282 

-1.2730044 

-0.20741354 

165 

+0.00001044 

-1.7023416 

-0.20114407 

-0.12716787 

-0.6051233 

-0.27625376 

180 

+0.13212520 

-1.5390947 

-0.22013939 

-0.10126618 

0.0000000 

-0.30227222 

195 

+  0.29405223 

-1.3852923 

-0.21561000 

-0.07503510 

+0.4924233 

-0.29612150 

210 

+0.44259806 

-1.2520945 

-0.19638955 

-0.05218717 

+0.8604054 

-0.26990714 

225 

+0.53811675 

-1.1428753 

-0.16935246 

-0.03375802 

+  1.1118587 

-0.23300075 

240 

+0.55512295 

-1.0569737 

-0.13887026 

-0.01942443 

+  1.2611694 

-0.19133211 

255 

+0.48911303 

-0.9922445 

-0.10732663 

-0.00839923 

+  1.3226839 

-0.14811570 

270 

+0.35776744 

-0.9463685 

-0.07579537 

+0.00013574 

+  1.3083458 

-0.10478640 

285 

+0.19621617 

-0.9174587 

-0.04456334 

+  0.00686904 

+  1.2273328 

-0.06171770 

300 

+0.04762640 

-0.9042927 

-0.01348260 

+0.01232297 

+  1.0864013 

-0.01870356 

315 

-0.04835263 

-0.9063942 

+0.01780841 

+0.01685710 

+0.8903706 

+0.02473967 

330 

-0.06620421 

-0.9240677 

+0.04973149 

+0.02067163 

+0.6425655 

+0.06916321 

345 

-0.00136711 

-0.9584216 

+0.08262385 

+0.02378861 

+0.3452194 

+0.11498675 

V 

*-l 

+  2.90467772 

-5.9084737 

+0.01066288 

-0.40860954 

-3.5384806 

+0.02421404 

2* 

+2.90467759 

-5.9084840 

+0.01066372 

-0.40860959 

-3.5384837 

+0.02421607 

sin  <p  • 


!(">  +  cos  <p  •  B0(c)  =  +  0.000000033. 


98 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  MERCURY  ON  VENCJS. 


—  Ro  COS  V 


E 

[«o  sin  v 

L 

(            \ 

,     Wo  cos  u 

Wo  sin  u 

-2-flo 

+  (cos  v+cos  E)S( 

a860  *"   / 

vSo 

a 

0* 

+0.2330866 

+  1.0113711 

+0.015230839 

+0.021064164 

2.0089005 

15 

+0.0087244 

+  1.1264720 

+0.009475283 

+0.024986282 

2.1567748 

30 

-0.2767659 

+  1.2080644 

+  0.002459142 

+0.024755972 

2.3539076 

45 

-0.6298334 

+  1.2226121 

-0.003043897 

+0.018369962 

2.6050539 

60 

-1.0391050 

+  1.1217558 

-0.002204101 

+0.004843036 

2.9088638 

75 

-1.4586399 

+0.8508163 

+0.011332120 

-0.013742297 

3.2495836 

90 

-1.7948636 

+0.3765263 

+0.042079553 

-0.029990049 

3.5871504 

105 

-1.9215248 

-0.2668018 

+0.086138100 

-0.032190511 

3.8554443 

120 

-1.7441606 

-0.9487539 

+0.127283754 

-0.012324508 

3.9830138 

135 

-1.2790794 

-1.4880990 

+0.145357191 

+0.024082111 

3.9330324 

150 

-0.6591294 

-1.7579567 

+0.132584634 

+0.059936528 

3.7270060 

165 

-0.0490234 

-1.7488888 

+0.098503636 

+0.080428204 

3.4271881 

180 

+0.4402788 

-1.5390947 

+0.059336027 

+0.082061396 

3.0992536 

195 

+0.7728021 

-1.2274791 

+0.026854296 

+0.070065081 

2.7888981 

210 

+0.9628351 

-0.8906639 

+0.005513895 

+0.051895060 

2.5190296 

225 

+  1.0443017 

-0.5731024 

-0.005195888 

+0.033355748 

2.2968106 

240 

+  1.0518039 

-0.2937729 

-0.007836004 

+0.017773731 

2.1211810 

255 

+  1.0129579 

-0.0559799 

-0.005257515 

-0.006550236 

1.9880041 

270 

+0.9468650 

+0.1451147 

+0.000109448 

-0.000080285 

1.8927371 

285 

+0.8649660 

+0.3177537 

+0.006402057 

-0.002489441 

1.8316679 

300 

+0.7723975 

+0.4708818 

+0.012250672 

-0.001332937 

1.8023971 

315 

+0.6691423 

+0.6125550 

+0.016656295 

+0.002594179 

1.8040170 

330 

+0.5508294 

+0.7487960 

-f  0.018894170 

+0.008386104 

1.8371831 

345 

+0.4092814 

+0.8824106 

+0.018479728 

+0.014979907 

1.9041728 

2, 

-0.5559282 

-0.3477320 

+0.405702029 

+0.226988212 

31.8406236 

22 

-0.5559251 

-0.3477313 

+0.405701406 

+0.226989461 

31.8406376 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[de/dt]oo  =  - 

97592.111m'  n4 

.9894147 

[dxldtiw  =  -8920493.9   TO'  n  6 

.9503889 

[di/dt]M  =  + 

71223.820  TO'  p4 

.8526253 

[dQ/(ft]oo  =  + 

673299.06  TO'  p5 

.8282080 

[dirldt}w  =  -8919313.6   TO'  »6 

.9503315 

[dL/dt]oo  =  +5590689.3   m'  p  6 

.7474654 

OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  m'. 

[de/d/Joo    =  -0.013012279 

=  -1.1893992 

=  +0.0094965089 

=  +0.089773204 

=  -1.1892420 
[dL/dt]w   =  +0.74542525 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


[de/dt]^    -0.01304 

-0.01301 

-0.013012 

e[dir/dt]oo    -0.00810 

-0.00814 

-0.008138 

[di/dt}^    +0.00950 

+0.00949 

+0.009497 

sin  i  [dfl/d<]oo    +0-00529 

+0.00531 

+0.005301 

[dL/dfloo   +0.747 

+0.745425 

NOTES. 

This  computation  was  originally  made  with  but  twelve  points  of  division,  but 
it  was  found  that,  notwithstanding  the  small  eccentricity  of  the  orbit  of  Venus,  the 
values  of  e'  and  I  are  here  so  large  that  the  tests  which  arise  by  comparing  the  sums 
of  the  functions  were,  toward  the  close  of  the  computation,  entirely  inapplicable. 
The  sums  for  [de/dt\00  agreed  to  but  a  single  significant  figure,  while  those  for  [dx/dt]0o, 
[di/dt]00,  and  [dQ/di\00  agreed  to  but  two.  It  will  be  noticed  that  the  increase  of  the 
number  of  points  of  division  almost  wholly  removes  the  discrepancy. 

Notwithstanding  the  entire  disagreement  of  the  test  equations  when  but  twelve 
points  of  division  were  employed,  it  is  evident  that  this  number  would  have  been 
sufficient.  The  greatest  error  would  have  occurred  with  [dLjdt]OQ,  its  amount  being 
0".00000016,  showing  that  with  this  coefficient  the  sum  of  all  terms  of  an  order 
higher  than  the  12th  amounts  to  but  1 /4000000th  of  the  remaining  terms. 


100 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


.E 

0° 

30 

60 

90 

120 

150 

180 

210 

240 

270 

300 

330 


A 

1.53711333 
1.52932317 
1.51985488 
1.51125435 
1.50583081 
1.50503256 
1.50906482 
1.51684286 
1.52628659 
1.53487476 
1.54031063 
1.54113325 

9.13846106* 
9.13846095 


ACTION  OF  THE 

EARTH  ON  VENUS. 

B  cos  t 

B  am  t 

1000  Xff 

+0.64403672 

+0.34841048 

0.03414346 

+0.38337105 

+0.61703357 

0.10708849 

+0.02331740 

+0.71967967 

0.14568124 

-0.33964844 

+0.62884478 

0.11122751 

-0.60827014 

+0.36886855 

0.03827088 

-0.71057033 

+0.00941083 

0.00002491 

-0.61913831 

-0.35321171 

0.03509098 

-0.35847281 

-0.62183471 

0.10876150 

+0.00158081 

-0.72448100 

0.14763156 

+0.36454672 

-0.63364618 

0.11293247 

+0.63316838 

-0.37366983 

0.03927365 

+0.73546881 

-0.01421211 

0.00005681 

+0.07469486f 

-0.01440384t 

0.44009177 

+0.07469500 

-0.01440382 

0.44009169 

1.00234463 
1.00372554 
1.00314070 
1.00120600 
0.99983131 
1.0003553T 
1.00226402 
1.00370935 
1.00326747 
1.00133527 
0.99985008 
1.00036791 

6.01069819 
6.01069936 


E 


G 


G' 


G" 


0 

0.53448743 

1.00227181 

0.53462398 

0.00006372 

46 

55 

3.02 

30 

0.52531637 

1.00350237 

0.52574251 

0.00020298 

46 

22 

33.07 

60 

0.51643292 

1.00284204 

0.51701255 

0.00028098 

45 

53 

55.40 

90 

0.50976708 

1.00097979 

0.51021108 

0.00021779 

45 

33 

45.30 

120 

0.50571824 

0.99975382 

0.50587139 

0.00007567 

45 

20 

44.51 

150 

0.50439599 

1.00035525 

0.50439608 

0.00000005 

45 

14 

29.83 

180 

0.50651954 

1.00219338 

0.50665928 

0.00006911 

45 

19 

11.90 

210 

0.51285225 

1.00348844 

0.51328431 

0.00021116 

45 

39 

53.44 

240 

0.52273786 

1.00296095 

0.52332564 

0.00028127 

46 

15 

21.69 

270 

0.53325823 

1.00109414 

0.53371072 

0.00021137 

46 

54 

18.41 

300 

0.54017928 

0.99976461 

0.54033745 

0.00007270 

47 

19 

22.49 

330 

0.54048407 

1.00036779 

0.54048430 

0.00000011 

47 

18 

38.53 

2, 

3.12607525 

6.00978661 

3.12783029 

0.00084345 

277 

3 

39.01 

22 

3.12607397 

6.00978778 

3.12782900 

0.00084346 

277 

3 

38.58 

*  6as  +  3aV  +  6[o'!  -  2fcaa'ee'  cos  A']  =  9.13846101. 
t  6[a'V  -  koa'e  cos  A']  =  +  0.07469471. 
I  -  6k'aa'  cos  <p'  •  e  sin  A'  =  -  0.04440383. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


101 


ACTION 

OF 

THE  EARTH  ON 

VENUS. 

E 

logtfo 

log  Lo' 

logJVo 

\ogN 

logP 

logQ 

0° 

q 

.26000336 

0.60598257 

0.54577154 

9 

.8305330 

0.4344894 

0.3752914 

30 

o 

.25288171 

0 

.59719127 

0.53614961 

9.8233230 

0.4173018 

0.3578664 

60 

0 

.24673133 

0.58958546 

0. 

52781896 

9 

.8197364 

0.4066135 

0.3462012 

90 

0 

.24246622 

0.58430371 

0. 

52203035 

9.8196999 

0.4029640 

0.3412105 

120 

0 

.23974361 

0.58092896 

0.51833025 

9.8208350  • 

0.4019120 

0.3392393 

150 

0 

.23844519 

0.57931865 

0. 

51656434 

9 

.8213596 

0.4003696 

0.3377696 

180 

0 

.23942219 

0.58053038 

0. 

51789317 

9.8218870 

0.4004544 

0.3387987 

210 

0 

.24375790 

0.58590391 

0. 

52378440 

9.8244982 

0.4071947 

0.3466789 

240 

0 

.25132635 

0.59526901 

0. 

53404476 

9.8301980 

0.4226554 

0.3628370 

270 

0 

.25983876 

0.60577958 

0. 

54554439 

9.8370023 

0.4416487 

0.3819851 

300 

0 

.26543283 

0.61267386 

0. 

55308969 

9 

.8405753 

0.4533906 

0.3937357 

330 

0 

.26526804 

0.61247091 

0. 

55286781 

!) 

.8378789 

0.4500302 

0.3905869 

2,  . 

1 

.50265967 

3.56497024 

3. 

19694837 

8.9637646 

2.5195152 

2.1561032 

22 

1 

.50265782 

3.56496803 

3. 

19694594 

8.9637618 

2.5195089 

2.1560973 

E 

logF 

j. 

J, 

J 

i 

F, 

0° 

0.3752596 

0.9974825 

+0.004202020 

-0.047236424 

-0.005832177 

30 

0.3577653 

0.9964541 

+0.010025286 

-0.058127293 

-0 

.010328762 

60 

0.3460609 

0.9970986 

+0.013405599 

-0.053273840 

-0.012046994 

90 

0.3411015 

0.9987570 

+0.012206904 

-0.033976547 

-0.010526476 

120 

0.3392013 

0.9997616 

+0.006516023 

-0.005406098 

-0 

.006174632 

150 

0.3377695 

0.9991244 

-0.001172845 

+0.024782063 

-0 

.000157532 

180 

0.3387641 

0.9974877 

-0.007608410 

+0.048499056 

+0 

.005912547 

210 

0.3465734 

0.9964670 

-0.010818504 

+0.059389945 

+0.010409132 

240 

0.3626968 

0.9970681 

-0.010873469 

+0.054536468 

+0.012127366 

270 

0.3818798 

0.9987050 

-0.008962465 

+0.035239146 

+0.010606846 

300 

0.3936994 

0.9997505 

-0.005884342 

+0.006668715 

+0.006255003 

330 

0.3905868 

0.9991427 

-0.001520954 

-0.023519435 

+0.000237902 

2, 

2.1556821 

5.9886490* 

-0.000242579 

+0.003787877 

+0 

.000241113 

22 

2.1556763 

5.9886502 

-0.000242578 

+0.003787879 

+0 

.000241110 

*  • 

-  1  (t) 

r/  _  G")  = 

5.9878055. 

-  ^  («J 

Y  _  G")  .= 

5.9878067. 

102 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  THE  EARTH  ON  VENUS. 


E 

1000  X  F, 

So 

1000  X  So 

TF0 

B<" 

S<"> 

0° 

+0.20122394 

0.42126107 

-5.890097 

-0.11153496 

0.0000000 

-0.008199111 

30 

+0.06017487 

0.40675651 

-4.150418 

-0.13232119 

+0.2828447 

-0.005772122 

60 

-0.29485277 

0.40441131 

-0.984602 

-0.11894044 

+0.4858527 

-0.001365876 

90 

-0.50823894 

0.41101802 

+0.151227 

-0.07580674 

+0.5682286 

+0.000209069 

120 

-0.36563277 

0.41646740 

-1.349097 

-0.01272800 

+0.4969244 

-0.001858754 

150 

-0.00856116 

0.41403810 

-2.948800 

+0.05391799 

+0.2845157 

-0.004052670 

180 

+0.20680805 

0.40710579 

-1.730893 

+0.10632289 

0.0000000 

-0.002376680 

210 

+0.06559223 

0.40536093 

+2.554045 

+0.13208012 

-0.2785530 

+0.003510140 

240 

-0.29105383 

0.41526071 

+7.029019 

+0.12494379 

-0.4954847 

+0.009684416 

270 

-0.50707628 

0.43289840 

+7.732079 

+0.08349760 

-0.5984781 

+0.010689527 

300 

-0.36741788 

0.44451245 

+3.199250 

+0.01546615 

-0.5340294 

+0.004438118 

330 

-0.01281582 

0.43893384 

-3.068000 

-0.05784756 

-0.3052198 

-0.004266768 

s, 

-0.91092526 

2.50901873 

+0.273580 

+0.00352943 

-0.0467370 

+0.000322113 

S2 

-0.91092510 

2.50900580 

+0.270133 

+0.00352022 

-0.0466619 

+0.000317176 

_  [fl0  sin  v 

+  (coav+coaE)Sa] 

0°    -0.01178019 

30  +0.19740436 

60  +0.35044516 

90  +0.41100745 

120  +0.36078900 

150  +0.21090709 

180  +0.00346179 

210     -0.20590981 

240     -0.36545661 

270     -0.43294230 

300     -0.38308894 

330     -0.22607885 

S,     -0.04562979 
22     -0.04561206 

sin  <f\Ai(''  +  coa  <fB 


— 

- 


Ro cos  v 

.('-sec?    +\\smvsl 
\osee  *"      /S"       °J 

-0.42126107 
-0.35572418 
-0.20183130 
+0.00311509 
+0.20803111 
+0.35633148 
+0.40710579 
+0.34919573 
+0.19760050 
-0.01250179 
-0.22551772 
-0.37629530 

-0.03587269 
-0.03587897 

=  +  0.00000000814. 


TF.COSM 


Wt,  sin  u 


-2-flo 
a 


-0.065352939 

-0.09038273 

-0.83675673 

-0.013079792 

-0.13167315 

-0.80869188 

+0.049268295 

-0.10825646 

-0.80605518 

+0.061732775 

-0.04399687 

-0.82203004 

+0.012668751 

-0.00122668 

-0.83578481 

-0.049130989 

-0.02221027 

-0.83298365 

-0.062298971 

-0.08615914 

-0.81978340 

-0.013955080 

-0.13134085 

-0.81552660 

+0.050403535 

-0.11432600 

-0.83336328 

+0.067326200 

-0.04938656 

-0.86579680 

+0.015375410 

-0.00167292 

-0.88598306 

-0.052873488 

-0.02346769 

-0.87266531 

+0.000064081 

-0.40202393 

-5.01772646 

+0.000019626 

-0.40207539 

-5.01770028 

OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  103 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[de/dt]w  =  -  16017!410w'  n  4.2045923 
[dx/dtlw  =  -1840673.3  m'  n  6.2649767 
[di/dt]00  =  +  14.695  m'  p  1.1671802 

[dQ/<ft]M  =  -2385136.3  TO'  n  6.3775132 
[ArAftJoo  =  -1844854.1  TO'  n  6.2659621 
[dL/dt]m  =  -1765973.3  m'  n  6.2469841 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 

[de/dt]w  =  -0^04898290 

[dx/dt}w  =  -5.6289701 

[di/dt]m  =  +0.000044940 

[dn/dt}^  =  -7.293993 

[dw/dt]m  =  -5.6417558 

[dL/dt]m  =  -5.4005288 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


[de/dt]oo 

-0.04875 

-0.04896 

-0.048982 

e[dTr/dt]oo 

-0.03873 

-0.03852 

-0.038607 

[dildt]oo 

+0.00006 

+0.00004 

+0.0000449 

sin  i  [dtt/dt]w 

-0.43154 

-0.43169 

-0.431698 

[dL/dt]m 

-5.397 

-5.4005 

NOTES. 

The  close  agreement  of  the  sums  of  the  functions  toward  the  beginning  of  the 
computation  is  here  caused  by  the  smallness  of  the  term  a'V;  the  ratio  of  the  major 
axes  is,  however,  so  large  that  the  expansion  of  the  perturbing  function  is  not  very 
rapidly  convergent.  The  greatest  error  arising  from  a  division  of  the  orbit  into  but 
six  parts  would  here  occur  with  the  coefficient  [dttjdiloa,  its  amount  being  0".0004, 
which  is  l/16000th  of  the  whole. 


104 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF 

E 

A 

B  cos  e 

0° 

2.6510232 

-0.78427108 

30 

2.6356206 

-0.87195528 

60 

2.6766222 

-0.66613155 

90 

2.7630522 

-0.22195070 

120 

2.8717556 

+0.34156979 

150 

2.9736020 

+0.87343460 

180 

3.0412895 

+  1.23113141 

210 

3.0566810 

+  1.31881567 

240 

3.0156549 

+  1.11299237 

270 

2.9292114 

+0.66881112 

300 

2.8205208 

+0.10529076 

330 

2.7187000 

-0.42657423 

2, 

17.0768662* 

+  1.34058170f 

2« 

17.0768672 

+  1.34058118 

E 

I 

G 

O 

0 

0.32836475 

2.30159781 

30 

0.31341820 

2.30194311 

60 

0.35238060 

2.30214618 

90 

0.43608420 

2.30201186 

120 

0.54393270 

2.30165423 

150 

0.64776765 

2.30144287 

180 

0.71834405 

2.30164190 

210 

0.73432570 

2.30208690 

240 

0.69089570 

2.30230785 

270 

0.60177725 

2.30207736 

300 

0.49293070 

2..30165418 

330 

0.39347570 

2.30144156 

2, 

3.12684850 

13.81100215 

2, 

3.12684870 

13.81100366 

*  6a2  +  3aV  +  6[a'2  - 

Zkaa'ee'  cos  K]  =  17.076! 

t  6[a'V  -  kaa'e  cos  K] 

=  +  1.34058156. 

t  —  6fc'aa'  cos  <p'  •  e  sin 

K'  =  -  0.0182328. 

MARS  ON  VENUS. 

/.'  sin  . 

+0.4410282 
-0.1200582 
-0.6497891 
-1.0062238 
-1.0938564 
-0.8892048 
-0.4471057 
+0.1139805 
+0.6437115 
+  1.0001462 
+  1.0877783 
+0.8831271 

-0.0182332J 
-0.0182329 


G' 

0.33433455 
0.31388465 
0.36444375 
0.46014840 
0.56837725 
0.66243750 
0.72188175 
0.73455360 
0.69835610 
0.62106785 
0.51868770 
0.41360955 

3.20608110 
3.20570155 


0.003928189 
0.000291101 
0.008527164 
0.020447915 
0.024164644 
0.015968474 
0.004037199 
0.000262374 
0.008368400 
0.020201647 
0.023896846 
0.015750931 

0.072922442 
0.072922442 


G" 

0.00510484 
0.00040288 
0.01016345 
0.01930383 
0.01847156 
0.01047414 
0.00242983 
0.00015516 
0.00520478 
0.01412953 
0.02001680 
0.01654686 

0.06139126 
0.06101240 


2.30246275 
2.30200670 
2.30404590 
2.30677230 
2.30762720 
2.30563865 
2.30274975 
2.30215960 
2.30456350 
2.30723845 
2.30739440 
2.30502860 

13.82884350 
13.82884430 


o  / 

22  33 
21  40 

23  41 

27  1 

30  11 

32  38 
34  6 
34  23 

33  30 

31  34 

28  47 
25  31 

172  54 

172  51 


26.00 
58.40 

4.25 
51.11 
38.49 
59.04 
10.10 
46.69 
59.53 
45.83 
46.93 

1.93 

5.30 
23.00 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


105 


ACTION  OF  MARS  ON  VENUS. 

E 

log  A!. 

log  ZV 

log  JVo         log  N         log  /' 

logQ 

0° 

0.05236235 

0.34221622 

0.25376720    9.0799241    8.6961571 

8.9706997 

30 

0.04824440 

0.36381569 

0.24772073    9.0778391     8.6903136 

8.9633892 

60 

0.05820469 

0.34986592 

0.26232763    9.0871721    8.7089460 

8.9854537 

90 

0.07644597 

0.37365893 

0.28892064    9.1058581    8.7480486 

9.0290445 

120 

0.09671054 

0.39993150 

0.31822634    9.1294237    8.7983322 

9.0821385 

150 

0.11444649 

0.42279102 

0.34867356    9.1516340    9.8464806 

9.1313354 

180 

0.12581914 

0.43738414 

0.35989311    9.1660121    8.8784042 

9.1634092 

210 

0.12819869 

0.44043118 

0.36327720    9.1681182    8.8842476 

9.1692445 

240 

0.12114959 

0.43139835 

0.35324262    9.1574139    8.8625240 

9.1475124 

270 

0.10649224 

0.41255431 

0.33228403    9.1373395    8.8203390 

9.1048461 

300 

0.08740987 

0.38789402 

0.30480686    9.1137453    8.7700379 

9.0527515 

330 

0.06771795 

0.36229172 

0.27622192    9.0929018    8.7249711 

9.0040125 

2, 

0.54165618 

2.34869015 

1.85226376    4.7336910    2.7144014 

4.4019649 

2a 

0.54154574 

2.34854285 

1.85209808    4.7336907    2.7144005 

4.4018722 

E 

logF 

Ji' 

J2             J, 

Ft 

0° 

S.9695199 

2.3059508 

+0.040158678     -0.039429682 

-0.09502735 

30 

8.9632959 

2.2999322 

-0.012465787     -0.068607310 

+0.02586866 

60 

8.9831123 

2.3089841 

-0.060962061     -0.080069400 

+0.14000859 

90 

9.0246325 

2.3187331 

-0.093160848     -0.070744754 

+0.21680880 

120 

9.0779450 

2.3192082 

-0.101285499     -0.043131936 

+0.23569076 

150 

9.1289682 

2.3119133 

-0.083189540     -0.004629729 

+0.19159498 

180 

9.1628614 

2.3032758 

-0.042922865     +0.034445271 

+0.09633687 

210 

9.1692095 

2.2997005 

+0.009559512     +0.063622976 

-0.02455913 

240 

9.1463381 

2.3040271 

+0.060251438     +0.075085208 

-0.13869908 

270 

9.1016472 

2.3135287 

+0.094787774     +0.065760616 

-0.21549926 

300 

9.0481938 

2.3207258 

+0.103054621     +0.038147729 

-0.23438118 

330 

9.0002162 

2.3179852 

+0.082763214     -0.000354612 

-0.19028541 

Si 

4.3879705 

13.8621718* 

-0.001705675     -0.014952810 

+0.00392861 

22 

4.3879695 

13.8617930 

-0.001705688     -0.014952813 

+0.00392864 

*2,(J,'-G")  =  13 

.800780G. 

SzG/i'  -  G")  =  13 

.8007806. 

106 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  MAKS  ON  VENUS. 

E 

1000  X  F  s 

Ro        1000  X  -So 

1000  X  Wo 

fiw 

1000  X  <S<"> 

0° 

+  2.8021516 

0.06350844   -0.9770071 

-  3.5365233 

0.00000000 

-1.3600097 

30 

-  0.4518739 

0.06287361   +0.1223535 

-  6.3268753 

+0.04372018 

+0.1701610 

60 

-  0.0998524 

0.06461988   +1.2993923 

-  7.7066718 

+0.07763320 

+  1.8025651 

90 

+  3.5312016 

0.06860530   +2.2776450 

-  7.2896438 

+0.09484620 

+3.1488232 

120 

+  6.8298095 

0.07412522   +2.6943730 

•  4.7318401 

+0.08844541 

+3.7122436 

150 

+  6.5062493 

0.07970667   +2.2590820 

-  0.1661617 

+0.05477227 

+3.1047600 

180 

+  2.8799133 

0.08340064   +1.0358414 

+  5.2294311 

0.00000000 

+  1.4223079 

210 

-  0.4389788 

0.08380114   -0.4699266 

+  9.3597858 

-0.05758587 

-0.6458416 

240 

-  0.1552801 

0.08095204    -1.6671420 

+  10.5057004 

-0.09659109 

-2.2969487 

270 

+  3.4223024 

0.07610223   -2.2705920 

+  8.5365166 

-0.10521063 

-3.1390719 

300 

+  6.6966141 

0.07080065   -2.2876620 

+  4.6568417 

-0.08505867 

-3.1735299 

330 

+  6.3844559 

0.06631737   -1.8208410 

+  0.3034394 

-0.04611486 

-2.5323023 

Zi 

+  18.9533560 

0.43740687   -0.0977956 

+  4.4169380 

-0.01557115 

+0.1066283 

S2 

+  18.9533565 

0.43740632   -0.0977209 

+  4.4170612 

-0.01557271 

+0.1065284 

\Ro  sin  v 

1  —Ro  cos  v 

E 

1     (T         \       n  1000 

X  W0  cos  u    1000  X  W0  sin  u 

-2-flo 

+  (COSv+COSE)S( 

+  1  sec2  <f  +  1  1  sin  vSo  \ 
\i       /      J 

a 

0° 

-0.001954014 

-0.06350844 

2.072195 

-  2.865834 

-0.12614768 

30 

+0.031835191 

-0.05421922 

0.625404 

-  6.295890 

-0.12500201 

60 

+0.057445987 

-0.02972268     + 

3.192309 

-  7.014411 

-0.12879747 

90 

+0.068588125 

+0.00502476     + 

5.936279 

•  4.230780 

-0.13721060 

120 

+0.061265775 

+0.04210058     + 

4.709813 

-  0.456037 

-0.14875771 

150 

+0.035700934 

+0.07141598     + 

0.151409 

+  0.068446 

-0.16035811 

180 

-0.002071682 

+0.08340064 

3.064140 

-  4.237688 

-0.16794274 

210 

-0.040838000 

+  0.07318497 

0.988919 

-  9.307396 

-0.16859554 

240 

-0.068190156 

+0.04377273     + 

4.238101 

-  9.612920 

-0.16245809 

270 

-0.076084930 

+0.00506196     + 

6.883205 

-  5.049116 

-0.15220446 

300 

-0.063800118 

-0.03106656     + 

4.629519 

-  0.503716 

-0.14111682 

330 

-0.036506252 

-0.05549212     + 

0.277348 

+  0.123100 

-0.13184872 

Si 

-0.017304208 

+0.04497627     +11.633407 

-24.690606 

-0.87522051 

22 

-0.017304932 

+0.04497633     +11.633918 

-24.691636 

-0.87521944 

sin  if  •  J.AI'"  +  cos  <f 

•  B0(c)  =  +  0.0000000018. 

OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  107 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

=  -         6075.5972  m'    n  3.7835890 
=  +2307588.8        m'    p  6.3631584 
[difdt]m    =  +       4084.7434m'    p  3.6111648 
[dQ/dfloo    =  -     146478.61      m'    n  5.1657742 
[dw/dt]w    =  +2307332.0       m'    p  6.3631101 
=  -  307497.75      m'    n  5.4878419 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  m'. 

[de/dt]w  =  -0.0019639882 

[dx/dt}00  =  +0.74594759 

[dt'/dfloo  =  +0.0013204280 

[dfl/dflw  =  -0.047350446 

[d7r/d<]oo  =  +0.74586465 

[dL/dt]w  =  -0.099401232 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


sn 


[de/dt]w 

-0^00195 

-0.00196 

-0.001964 

[dw/dilw, 

+0.00510 

+0.00510 

+0.005104 

[di/dtlw 

+0.00131 

+0.00132 

+0.001320 

[dtt/dt]w 

-0.00280 

-0.00281 

-0.002802 

[dLfdt]w 

-0.099 

-0.099401 

NOTES. 

The  close  agreement  of  the  final  sums  shows  that,  notwithstanding  the  high 
eccentricity  of  the  orbit  of  Mars,  the  expansion  of  the  perturbing  function  is  quite 
rapidly  convergent  for  this  case.  The  greatest  error  arising  from  a  division  into  but 
six  parts  would  here  occur  with  the  coefficient  [dQJdt]^,  and  would  amount  to 
1 /50000th  of  the  whole  value  of  this  coefficient. 


108 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


E  A 

0°  27.41848845 

30  27.28099847 

60  27.22724715 

90  27.27164617 

120  27.40230326 

150  27.58420446 

180  27.76860046 

210  27.90607834 

240  27.95980514 

270  27.91539376 

300  27.78474891 

330  27.60287233 

Zi  165.56119337* 

22  165.56119353 


E  I 

0  0.347678 

30  0.210701 

60  0.157583 

90  0.202099 

120  0.332232 

150  0.513472 

180  0.697718 

210  0.835716 

240  0.890108 

270  0.845842 

300  0.714692 

330  0.532182 

Zi  3.140010 

22  3.140011 


t  6[a'V  -  kaa'e  cos  K]  =  +  7.9087800. 
|  -  Gfc'aa'  cos  <p'  •  e  sin  K'  =  —  0.1367433, 


ACTION  OF  JUPITER  ON  VENUS. 

B  COS  e 

B  sin  t 

o 

h 

-0.4215334 

+3.3076466 

0.68960365 

27.007779 

-1.8560261 

+  1.9912702 

0.24993195 

27.007266 

-2.4400062 

+0.1352278 

0.00115264 

27.006632 

-2.0169963 

-1.7631551 

0.19594869 

27.006516 

-0.7003432 

-3.1952096 

0.64351702 

27.007039 

+  1.1571588 

-3.7772161 

0.89930104 

27.007701 

+3.0577930 

-3.3532271 

0.70874081 

27.007851 

+4.4922856 

-2.0368516 

0.26150512 

27.007330 

+5.0762663 

-0.1808092 

0.00206064 

27.006665 

+4.6532564 

+  1.7175743 

0.18594838 

27.006520 

+3.3366031 

+3.1496275 

0.62528757 

27.007025 

+  1.4791014 

+3.7316353 

0.87772776 

27.007658 

+7.9087796f 

-0.13674401 

2.67036233 

162.042990 

+7.9087799 

-0.1367430 

2.67036294 

162.042990 

0 

G' 

G" 

e 

0      1      II 

27.0068207 

0.4107944 

0.06215867 

7 

35  44.57 

27.0069201 

0.2483150 

0.03726867 

5 

53  53.53 

27.0066304 

0.1578551 

0.00027037 

4 

23  18.41 

27.0062448 

0.2334496 

0.03108029 

5 

40  35.79 

27.0061457 

0.3936568 

0.06053126 

7 

26  34.62 

27.0064436 

0.5728576 

0.05812876 

8 

46  58.16 

27.0068530 

0.7344471 

0.03573164 

9 

42  56.80 

27.0069600 

0.8475114 

0.01142507 

10 

16  14.68 

27.0066621 

0.8901968 

0.00008571 

10 

27  38.97 

27.0062568 

0.8541659 

0.00806094 

10 

17  29.01 

27.0061444 

0.7465850 

0.03101258 

9 

45  50.06 

27.0064305 

0.5886244 

0.05521465 

8 

52  22.88 

162.0392563 

3.3335352 

0.18979023 

49 

22   3.43 

162.0392558 

3.3449238 

0.20117838 

49 

47  34.05 

a'ee'cosK]  =  165.5611934, 

OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


109 


ACTION 

OF  JUPITER  ON  VENUS. 

E 

log/fo 

log  Lo' 

log  No         log  N 

logP 

logQ 

0° 

0.00574768 

0.28065745 

0.18470221     7.4290891 

4.8448027 

6.1813194 

30 

0.00346017 

0.27761215 

0.18127767    7.4282000 

4.8416643 

6.1774038 

60 

0.00191340 

0.27555166 

0.17896019    7.4297378 

4.8423401 

6.1772233 

90 

0.00320448 

0.27727161 

0.18089469    7.4332724 

4.8466166 

6.1822034 

120 

0.00551791 

0.28035168 

0.18435840    7.4378460 

4.8533279 

6.1897695 

150 

0.00769513 

0.28324824 

0.18761506    7.4422392 

4.8606851 

6.1974532 

180 

0.00942826 

0.28555246 

0.19020526    7.4452930 

4.8667491 

6.2034501 

210 

0.01054435 

0.28703562 

0.19187230    7.4462011 

4.8699179 

6.2064140 

240 

0.01094130 

0.28756298 

0.19246500    7.4447133 

4.8693318 

6.205706T 

270 

0.01058711 

0.28709242 

0.19193614    7.4412095 

4.8651139 

6.2015516 

300 

0.00952253 

0.28567775 

0.19034610    7.4366174 

4.8583732 

6.1950025 

330 

0.00785489 

0.28346069 

0.18785389    7.4321742 

4.8509265 

6.1876739 

2i 

0.04307108 

1.69535398 

1.12103716    4.6232965 

9.1349247 

7.1524707 

Zi 

0.04334613 

1.69572073 

1.12144975    4.6232963 

9.1349242 

7.1526998 

E 

log  V 

Ji' 

J* 

J3 

Ft 

0° 

6.1800737 

27.032601604 

+0.14514043 

-0.9796507 

-4.3121624 

30 

6.1766560 

27.002428557 

+0.10192507 

-1.0502482 

-2.5960090 

60 

6.1772178 

26.980449138 

+0.02691302 

-0.8390983 

-0.1762960 

90 

6.1815797 

27.031356993 

-0.07055224 

-0.4027785 

+2.2986164 

120 

6.1885563 

27.065863046 

-0.15999110 

+0.1417995 

+4.1655781 

150 

6.1962891 

27.048572013 

-0.20267009 

+0.6487166 

+4.9243381 

180 

6.2027348 

27.006174576 

-0.17674090 

+0.9821448 

+4.3715859 

210 

6.2061853 

26.976509919 

-0.09316228 

+  1.0527424 

+  2.6554331 

240 

6.2057043 

26.979786076 

+0.01124440 

+0.8415929 

+0.2357201 

270 

6.2013902 

27.007935305 

+0.09774464 

+0.4052732 

-2.2391928 

300 

6.1943816 

27.036474318 

+0.11682513 

-0.1393049 

-4.1061538 

330 

6.1865682 

27.045934111 

+0.16010585 

-0.6462222 

-4.8649146 

Si 

7.1486685 

162.101348758*     -0.00660902 

+0.0074833 

+0.1782719 

2j 

7.1486684 

162.112736898 

-0.00660905 

+0.0074833 

+0.1782712 

*  z,(J,'  -  G")  = 

161.911558528. 

22(Ji'  -  C")  = 

161.911558518. 

110 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  JUPITEK  ON  VKNUS. 

E 

Ft 

1000  X  flo      100000  X  So      1000  X  W0      1000  X  «<">     100000  X  £<"' 

0° 

+0.06417143 

1.3439836   -0.81929943   -0.14785238     0.0000000    -1.1404782 

30 

-0.01379964 

1.3378723   -0.27202988   -0.15783804   +0.9303113   -0.3783208 

60 

-0.00425156 

1.3439282   +0.28211779   -0.12622122   +1.6145721   +0.3913643 

90 

+0.08385809 

1.3588647   +0.54292475   -0.06059607   +1.8786178   +0.7505883 

120 

+0.16287446 

1.3765538   +0.50194299   +0.02305118   +1.6424891   +0.6915652 

150 

+0.15397825 

1.3917093   +0.38822446   +0.10305711   +0.9563448   +0.5335546 

180 

+0.06595223 

1.4018961   +0.39768404   +0.15712801     0.0000000   +0.5460578 

210 

-0.01357095 

1.4066592   +0.47042587   +0.16914113   -0.9666180   +0.6465278 

240 

-0.00563629 

1.4053074   +0.35504098   +0.13510528    -1.6767973   +0.4891671 

270 

+0.08123098 

1.3964225   -0.08725072   +0.06503287   -1.9305409   -0.1206233 

300 

+0.15970886 

1.3800824   -0.66641223   -0.02064190   -1.6580071    -0.9244718 

330 

+0.15112247 

1.3602249   -0.99122256   -0.09822805   -0.9458544   -1.3785254 

2i 

+0.44281913 

8.2517515   +0.05107414   +0.02056897   -0.0777432   +0.0532044 

22 

+0.44281920 

8.2517529   +0.05107192   +0.02056895   -0.0777394   +0.0532012 

E 

1000  X[Ro  sin  v 

+  (cosv+cosE)S<>] 

1000  x|  -Ro  cost) 
,  1000  X  Wo  cos  u     1000  X  W0  sin  u         -  2  -  «0 

/;•         \  .       1                                                        (j 

~\-  1  -  SGC^  (f  -\~\  isin  v  *oo  I 

0° 

-0.0163860 

-1.3439836     -0.086632804     -0.11981268     -0.0026695736 

30 

+0.6682014 

-1.1590573     -0.015602101     -0.15706504     -0.0026598877 

60 

+  1.1706512 

-0.6601482     +0.052284193     -0.11488325     -0.0026786592 

90 

+  1.3587959 

+0.0201574     +0.049346057     -0.03516889     -0.0027177294 

120 

+  1.1829926 

+0.7039968     -0.022943878     +0.00222159     -0.0027625280 

150 

+0.6850081 

+  1.2114930     -0.093907380     -0.04245200     -0.0027999142 

180 

-0.0079537 

+  1.4018961     -0.092067787     -0.12732924     -0.0028229786 

210 

-0.7073256 

+  1.2159042     -0.017870798     -0.16819438     -0.0028299915 

240 

-1.2164221 

+0.7037025     +0.054502775     -0.12362395     -0.0028202318 

270 

-1.3963839 

+0.0113009     +0.052437627     -0.03846518     -0.0027928449 

300 

-1.2058917 

-0.6713715     -0.020520787     +0.00223277     -0.0027507212 

330 

-0.7013026 

-1.1657067     -0.089781837     -0.03984932     -0.0027043278 

2i 

-0.0930097 

+0.1340921     -0.115378288     -0.48119476     -0.0165046924 

22 

-0.0930067 

+0.1340915     -0.115378432     -0.48119481     -0.0165046855 

sin  (p  •  \A  i(<)  +  cos  (f  • 

Bo(c)  =  +  0.0000000000028. 

OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  Ill 

DIFFERENTIAL  COEFFICIENTS. 

u  log  coeff. 

[de/dt]M    =  -       32.654970  m'  n  1.5139493 

[dx/dt]m    =  +6879.8159      TO'  p  3.8375768 

[di/dt]M     =  -      40.510972  TO'  n  1.6075727 

[<&/#]«,    =  -2854.6599     TO'  n  3.4555544 

[dir/eft]oo    =  +6874.8117      TO'  p  3.8372608 

[dL/dt]m   =  -5799.7390      TO'  n  3.7634084 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 


=  -0.031162921 

[dxldt]<n    =  +6.5654682 
[di/dt]M    =  -0.038659982 
[dQ/dt]w    =  -2.7242270 
=  +6.5606924 
=  -5.5347410 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


-0.03117  -0.0311629 

+0.04482  +0.04491  +0.0448955 

[di/dt]M    -0.03865  -0.03865  -0.0386600 

sin  i  [daAft]M    -0.16114  -0.16122  -0.1612345 

[dL/dt]w   -5.535  -5.5347410 

NOTES. 

The  term  aV  is  here  so  large  that  the  sums  of  the  functions  B,  e,  G',  G",  and  0, 
as  well  as  those  of  the  functions  immediately  dependent  upon  these  quantities  are 
in  great  disagreement;  but,  as  the  expansion  of  the  perturbing  function  is  here  rapidly 
convergent,  the  final  sums  agree  almost  exactly.  The  greatest  effect  of  all  terms  from 
the  6th  to  the  llth  orders  is  here  produced  with  the  coefficient  [di/dt]^  and  amounts 
to  0".00000002. 


112 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  SATURN  ON  VENUS. 

E 

A 

B  cos  t 

B  sin  e 

g 

I 

0° 

92.09886822 

+  10.392665 

+4.3401600 

5.386574 

90.704833 

30 

91.77434432 

+  7.489741 

+6.4166597 

11.773870 

90.705340 

60 

91.37861335 

+  3.936959 

+6.7658078 

13.090030 

90.705178 

90 

91.01772022 

+  0.686282 

+5.2940524 

8.014522 

90.704527 

120 

90.78837022 

•  1.391274 

+2.3957451 

1.641279 

90.704015 

150 

90.75201312 

-  1.739027 

-1.1525126 

0.379833 

90.704174 

180 

90.91838168 

-  0.263800 

-4.3999687 

5.536055 

90.704848 

210 

91.24289345 

+  2.639122 

-6.4764687 

11.994380 

90.705369 

240 

91.63859982 

+  6.191904 

-6.8256188 

13.322485 

90.705222 

270 

91.99948069 

+  9.442580 

-5.3538604 

8.196630 

90.704548 

300 

92.22884291 

+  11.520135 

-2.4555545 

1.724250 

90.704026 

330 

92.26522467 

+  11.867891 

+  1.0927030 

0.341433 

90.704165 

S, 

549.05167620* 

+30.386589f 

-0.179429U 

40.700673 

544.228122 

•*•} 

549.05167647 

+30.386589 

-0.1794266 

40.700668 

544.228123 

E 

I 

G 

G' 

G" 

e 

0 

0 

i   a 

0 

+  1.108078 

90.7041702 

1.1599384 

0.0511977 

6 

38   1.200 

30 

+0.783048 

90.7038960 

0.9248448 

0.1403539 

6 

12  59.177 

60 

+0.387477 

90.7035801 

0.6213419 

0.2322660 

5 

33  35.601 

90 

+0.027235 

90.7035526 

0.3116929 

0.2834826 

4 

38  20.618 

120 

-0.201603 

90.7038159 

0.0673334 

0.2687363 

3 

29   4.472 

150 

-0.238118 

90.7041279 

0.0164527 

0.2545245 

3 

7  43.816 

180 

-0.072423 

90.7041756 

0.2137666 

0.2855179 

4 

14  53.293 

210 

+0.251567 

90.7039071 

0.5115373 

0.2585083 

5 

16  44.963 

240 

+0.647420 

90.7035910 

0.8267173 

0.1776657 

6 

2   4.051 

270 

+  1.008976 

90.7035405 

1.0926850 

0.0827021 

6 

32   0.533 

300 

+  1.238859 

90.7038135 

1.2542286 

0.0151565 

6 

47  36.301 

330 

+  1.275102 

90.7041229 

1.2780897 

0.0029452 

6 

49  30.415 

s, 

+3.107808 

.  544.2231463 

4.1433262 

1.0305401 

32 

45  14.918 

22 

+3.107809 

544.2231470 

4.1353023 

1.0225166 

32 

37  19.522 

*  6o2  +  3aV  +  6[a'2  - 

Zkaa'ee'  cos  K\  =  549.05167622. 

t  6[a'V  -  kaa'e  cos  K\ 

=  +  30.386587. 

J  —  6fc'aa'  cos  if'  •  e  sin 

K'  =  -  0.1794290. 

OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


113 


ACTION  OF  SATURN  ON  VENUS. 

E 

logtfo 

log  /V         log  No         log  N        log  P 

logQ 

0° 

0.00437971 

0.27883659     0.18265468     6.6396204     3.0027123 

4.8644028 

30 

0.00384466 

0.27812417     0.18185352     6.6392492     3.0007786 

4.8628053 

60 

0.00307377 

0.27709752    0.18069890    6.6400076    2.9996349 

4.8619714 

90 

0.00213852 

0.27585161    0.17929758    6.6416827    2.9995754 

4.8620009 

120 

0.00120578 

0.27460866    0.17789947    6.6438207    3.0006087 

4.862S099 

150 

0.00097200 

0.27429705     0.17754895     6.6458518     3.0024609 

4.8645568 

180 

0.00179285 

0.27539102    0.17877951     6.6472415    3.0046480 

4.8670287 

210 

0.00277061 

0.27669370    0.18024473    6.6476233    3.0065929 

4.8690060 

240 

0.00362228 

0.27782805    0.18152050    6.6468912    3.0077706 

4.8699374 

270 

0.00424801 

0.27866126     0.18245753     6.6452313     3.0078525 

4.8696688 

300 

0.00459387 

0.27912172     0.18297533     6.6430830     3.0068083 

4.8683601 

330 

0.00463699 

0.27917912     0.18303987     6.6410259     3.0049227 

4.8664247 

Si 

0.01866826 

1.66288356     1.08452839     9.8606643     8.0221827 

9.1945101 

Yo 

0.01861079 

1.66280691     1.08444218     9.8606642     8.0221829 

9.1944623 

E 

logF 

J,'             J2             J3 

F2 

0° 

4.8640969 

90.64262161      +0.26581387      -3.1529304 

-22.089684 

30 

4.8619670 

90.77821079     +0.41797337      -2.4126056 

-32.658241 

60 

4.8605842 

90.92403274      +0.41481456      -1.0167115 

-34.435270 

90 

4.8603076 

90.98237184      +0.27423234      +0.6607234 

-26.944615 

120 

4.8612040 

90.92121232     +0.07649557     +2.1702330 

-12.193388 

150 

4.8630356 

90.85355551      -0.10022240      +3.1073439 

+  5.865829 

180 

4.8653231 

90.87694181      -0.22507782      +3.2209585 

+22.394088 

210 

4.8674621 

90.89557561      -0.30595380     +2.4806343 

+  32.962649 

240 

4.8688763 

90.86858755      -0.34689500     +1.0847396 

+34.739680 

270 

4.8691748 

90.78219477      -0.32168268      -0.5926962 

+27.249019 

300 

4.8682696 

90.66900008      -0.19522740      -2.1022048 

+  12.497792 

330 

4.8664071 

90.60246403     +0.02557719      -3.0393154 

-  5.561423 

2j 

9.1883540 

544.90239611*     -0.01007622      +0.2040844 

+  0.913218 

22 

9.1883541 

544.89437255      -0.01007598      +0.2040844 

+  0.913218 

*  2,(J,'  -  G")  = 

543.87185601. 

22(J,'  -  G")  = 

543.87185600. 

114 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  SATURN  ON  VENUS. 


E 

F3        1000  X  Ro     100000  X  S0      1000  X  W»      1000  X  B(n) 

100000  X  <S<»> 

0° 

-0.15178108   0.21824849   -0.02789044   -0.023072725    0.00000000 

-0.03882394 

30 

-0.76855930   0.21811010   -0.02299898   -0.017634117   +0.15166640 

-0.03198543 

60 

-1.16657105   0.21852784   -0.04315220   -0.007491896   +0.26253554 

-0.05986233 

90 

-0.94659087   0.21909707   -0.07037836   +0.004695343   +0.30289965 

-0.09729742 

120 

-0.32871848   0.21963056   -0.06653491   +0.015732681   +0.26206084 

-0.09167023 

150 

+0.06775179   0.22023472   -0.01412236   +0.022675427   +0.15133936 

-0.01940900 

180 

-0.15599309   0.22103492   +0.06128516   +0.023605169    0.00000000 

+0.08415031 

210 

-0.77884429   0.22186276   +0.10918347   +0.018202966   -0.15245807 

+0.15005585 

240 

-1.18017310   0.22225231   +0.09717617   +0.007900358   -0.26518905 

+0.13388702 

270 

-0.95986467   0.22180036   +0.03944793   -0.004483114   -0.30663694 

+0.05453639 

300 

-0.33810816   0.22057741   -0.01719598   -0.015556236   -0.26499777 

-0.02385490 

330 

+0.06476239   0.21916657   -0.03744378   -0.022338577   -0.15240100 

-0.05207429 

2, 

-3.32134496   1.32027153   +0.00368780   +0.001117351   -0.00559044 

+0.00382593 

-"2 

-3.32134495   1.32027158   +0.00368792   +0.001117928   -0.00559070 

+0.00382610 

E 

m  v  [ff  ,'     100°  Xf-Socost; 
J   p                  T   1000  X  TFo  cos  u     1000  X  TFo  sin  u 

1000X-2-Ro 

1  +  ^seczY,+  lJsin!%Sol 

a 

0° 

-0.00055781     -0.21824849     -0.013519262     -0.018697064 

-0.43351010 

30 

+0.10930468     -0.18874417     -0.001743111     -0.017547753 

-0.43363510 

60 

+0.18946670     -0.10888722     +0.003103342     -0.006818927 

-0.43556030 

90 

+0.21909678    +0.00009174     -0.003823625     +0.002725091 

-0.43819414 

120 

+0.19022138     +0.10978821     -0.015659442     +0.001516257 

-0.44076418 

150 

+0.10971092     +0.19096268     -0.020662231     -0.009340619 

-0.44307990 

180 

-0.00122570     +0.22103492     -0.013831594     -0.019129016 

-0.44509495 

210 

-0.11216824     +0.19142744     -0.001923256     -0.018101079 

-0.44635520 

240 

-0.19279204     +0.11058265     +0.003187081     -0.007228982 

-0.44602556 

270 

-0.22179790     +0.00072885     -0.003614847     +0.002651640 

-0.44360072 

300 

-0.19184805     -0.10885437     -0.015464964     +0.001682669 

-0.43964535 

330 

-0.11088191     -0.18905102     -0.020417779     -0.009062352 

-0.43573540 

Si 

-0.00673552    +0.00541570     -0.052184839     -0.048675063 

-2.64060044 

22 

-0.00673567     +0.00541552     -0.052184849     -0.048675069 

-2.64060046 

sin  <f  •  Miw  +  cos  <p  •  Bow  =  +  0.00000000000029. 

OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  115 

DIFFERENTIAL  COEFFICIENTS. 


n 

log  coeff. 

[de/dt]w 

=  -     2.3648522  TO' 

n  0.3738040 

[dx/dt]w 

=  +277.85744     TO' 

p  2.4438220 

[di/dt]w 

18.322835    TO' 

n  1.2629927 

[dQ/dfloo 

=  -288.76199      TO' 

n  2.4605400 

[dir/dt]oo 

=  +277.35124     TO' 

p  2.4430301 

[dL/dt]w 

=  -927.63054      TO' 

n  2.9673751 

FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 

[de/dt]w    =  -0.00067536338 
[dxldt]M    =  +0.079351564 

=  -0.0052327048 

=  -0.082465731 

=  +0.079207000 
[dL/dt]w   =  -0.26491624 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


-0.00067 

-0.00067 

-0.00067536 

+0.00055 

+0.00054 

+0.00054202 

-0.00523 

-0.00523 

-0.00523270 

-0.00489 

-0.00488 

-0.00488077 

-0.265 

-0.26491624 

[di/dt]m 
sin  i  [dQ/dt}oo 
[dL/dt] 


NOTES. 

As  in  the  previous  case,  the  considerable  disagreement  of  the  sums  of  the  functions 
near  the  beginning  of  the  computation  nearly  disappears  as  the  work  progresses,  show- 
ing that  the  convergence  of  the  expansion  of  the  perturbing  function  is  here  very 
rapid.  The  greatest  error  which  would  have  arisen  from  the  neglect  of  all  terms  from 
the  6th  to  the  llth  orders  would  have  here  occurred  with  the  coefficient  [dx/dt]00  and 
would  have  amounted  to  1 /70000th  part  of  the  remaining  terms. 


116 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  URANUS  ON  VENUS. 

E 

A 

B  cos  ( 

/  >'  sin  i 

9 

i 

0° 

369.8294733 

28.016529 

-  8.627090 

60.35594 

367.496220 

30 

370.1021929 

30.912376 

-  2.059265 

3.43885 

367.496155 

60 

370.0319613 

30.136215 

+  5.076263 

20.89679 

367.496140 

90 

369.6376057 

25.896024 

+  10.867539 

95.77536 

367.496165 

120 

369.0247981 

19.327964 

+  13.762797 

153.60480 

367.496245 

150 

368.3577351 

12.191933 

+  12.986248 

136.75991 

367.496230 

180 

367.8151465 

6.400029 

+  8.745974 

62.03086 

367.496275 

210 

367.5424142 

3.504185 

+  2.178150 

3.84740 

367.496245 

240 

367.6126214 

4.280346 

-  4.957378 

19.92946 

367.496135 

270 

368.0069649 

8.520535 

-  10.748653 

93.69136 

367.496165 

300 

368.6197847 

15.088597 

-13.643909 

150.96247 

367.496240 

330 

369.2868724 

22.224622 

-12.867361 

134.26735 

367.496240 

^ 

2212.9337853* 

103.2496801 

+  0.356657} 

467.78032 

2204.977255 

^ 

2212.9337852 

103.249675 

+  0.356658 

467.78025 

2204.977200 

E 


G 


G' 


G" 


0 

0 

+  1.522310 

367.495771 

1.6238940 

0.1011370 

O 

3 

55 

40^93 

30 

+  1.795095 

367.496129 

1.8003166 

0.0051977 

4 

1 

9.46 

60 

+  1.724880 

367.495985 

1.7573874 

0.0323564 

4 

0 

5.50 

90 

+  1.330495 

367.495453 

1.5044398 

0.1732316 

3 

52 

23.80 

120 

+0.717605 

367.495105 

1.0990547 

0.3803068 

3 

38 

8.91 

150 

+0.050560 

367.495217 

0.6363646 

0.5847917 

3 

18 

7.21 

180 

-0.492075 

367.495816 

0.2329563 

0.7245710 

2 

55 

22.89 

210 

-0.764775 

367.496217 

0.0134531 

0.7782012 

2 

39 

26.72 

240 

-0.694455 

367.495988 

0.0708724 

0.7651843 

2 

43 

51.72 

270 

-0.300145 

367.495472 

0.3769271 

0.6763791 

3 

3 

57.85 

300 

+0.312600 

367.495121 

0.8167025 

0.5029833 

3 

25 

59.41 

330 

+0.979685 

367.495243 

1.2686692 

0.2879853 

3 

43 

48.64 

2, 

+3.090865 

2204.973786 

5.6008673 

2.5065388 

20 

39 

9.36 

22 

+3.090915 

2204.973731 

5.6001704 

2.5057866 

20 

38 

53.68 

*  6a2  +  3aV  +  6[a'2  -  2kaa'ee'  cos  K]  =  2212.9337852. 
t  6[a'V  -  kaa'e  cos  K]  =  +  103.249685. 
t  —  6fc'aa'  cos  <p'  •  e  sin  A"  =  +  0.356657. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


117 


ACTION  OF  UKANUS  ON  VENUS. 

E 

logtfo 

log!/         logJV0         log  AT         logP 

logQ 

0° 

0.00153256 

0.27504417    0.17838936    5.7255240    0.8698244 

3.3385415 

30 

0.00160469 

0.27514027     0.17849746     5.7265670     0.8711895 

3.3398056 

60 

0.00159052 

0.27512139    0.17847622    5.7286906    0.8732303 

3.3418760 

90 

0.00149009 

0.27498757    0.17832569    5.7313185    0.8753928 

3.3441876 

120 

0.00131281 

0.27475131    0.17805993    5.7337420    0.8770917 

3.3461011 

150 

0.00108264 

0.27444453    0.17771485    5.7353151    0.8778752 

3.3470878 

180 

0.00084825 

0.27413211    0.17736340    5.7356234    0.8775397 

3.3468789 

210 

0.00070103 

0.27393587    0.17714264    5.7345896    0.8761825 

3.3455607 

240 

0.00074042 

0.27398839    0.17720173    5.7324870    0.8741636 

3.3435328 

270 

0.00093336 

0.27424555    0.17749102    5.7298710    0.8720155 

3.3413115 

300 

0.00117044 

0.27456155    0.17784648    5.7274385    0.8703090 

3.3394395 

330 

0.00138191 

0.27484340    0.17816353    5.7258446    0.8695041 

3.3384162 

Si 

0.00719500 

1.64759892     1.06733712    4.3835054    5.2421585 

0.0563696 

22 

0.00719372 

1.64759719     1.06733519    4.3835056    5.2421594 

0.0563691 

E 

logF 

JY            J2            J, 

ft 

0° 

3.3383922 

367.09942052     -0.77238411      -13.377206 

+  148.77538 

30 

3.3397979 

366.74089875      -0.18040606      -16.560521 

+  35.51232 

60 

3.3418282 

366.88453611      +0.52246333      -15.273725 

-  87.54088 

90 

3.3439319 

367.40160988      +0.87586900      -  9.861620 

-187.41225 

120 

3.3455398 

367.86665960     +0.72768333      -  1.774376 

-237.34142 

150 

3.3462248 

367.95568703     +0.32642368  .   +  6.821041 

-223.94969 

180 

3.3458097 

367.72285452     +0.04295564     +13.621489 

-150.82559 

210 

3.3444123 

367.51508313      +0.01342241      +16.804806 

-  37.56251 

240 

3.3424036 

367.61069788     +0.04551264     +15.518012 

+  85.49071 

270 

3.3403134 

367.89469814      -0.13535069      +10.105905 

+  185.36209 

300 

3.3386972 

367.98729038      -0.54951786      +  2.018660 

+235.29114 

330 

3.3379911 

367.66272932      -0.88324571      -  6.576756 

+221.89949 

2i 

0.0526706 

2205.17145901*     +0.01671297     +  0.732854 

6.15066 

S2 

0.0526714 

2205.17070625     +0.01671263     +  0.732855 

6.15055 

*2,(J,'-G")  = 

2202.66492021. 

Zi(Ji'  -  G")  = 

2202.66491965. 

118 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  URANUS  ON  VENUS. 

E 

F3         1000  X  Ro     1000000  X  So    1000000  X  W0 

1000  X  fi<n) 

1000000  X  «S("> 

0° 

-  4.0226412   0.02650922   -0.05811066   -2.918780 

0.00000000 

-0.08089097 

30 

-  0.1770896   0.02652281   -0.01305234   -3.621493 

+0.01844306 

-0.01815231 

60 

•  1.6152186   0.02667654   +0.04940526   -3.356828 

+  0.03204873 

+0.06853681 

90 

-  6.9332675   0.02691093   +0.05269544   -2.182313 

+0.03720412 

+0.07285096 

120 

-10.8407275   0.02710189   -0.01759673   -0.401343 

+0.03233767 

-0.02424436 

150 

-  9.4434565   0.02715609   -0.09660941   +1.506695 

+0.01866093 

-0.13277471 

180 

-  4.1342719   0.02707758   -0.10424233   +3.017072 

0.00000000 

-0.14313458 

210 

-  0.2013093   0.02694790   -0.02527822   +3.713882 

-0.01851786 

-0.03474102 

240 

•  1.5455370   0.02684358   +0.07399797   +3.412656 

-0.03202946 

+0.10195266 

270 

-  6.7883563   0.02677475   +0.10841711   +2.207472 

-0.03701585 

+0.14988562 

300 

-10.6594103   0.02669990   +0.05468656   +0.432404 

-0.03207678 

+0.07586323 

330 

-  9.2743255   0.02659629   -0.02803268   -1.439065 

-0.01849416 

-0.03898596 

2! 

-32.8178065   0.16090871   -0.00185993   +0.185181 

+0.00028016 

-0.00191721 

2, 

-32.8178047   0.16090877   -0.00186010   +0.185178 

+0.00028024 

-0.00191742 

1000  X  f  Bo  sin  D  +    1000  X  [-  So  cos  v  + 

r 

E 

,          \   1    /r       \      1  100000°  X  ^o  C08  "  1000000  X  We  sin  u 
(cos  v  +  cos  E]  So  1   (~  secV  +  1  1  sin  vSo  \ 

1000  X  -2-  Ro 

0° 

-0.00011622      -0.02650922     -1.710234 

-  2.365243 

-0.05265561 

30 

+0.01331757      -0.02293687     -0.357980 

-  3.603757 

-0.05273125 

60 

+0.02323048      -0.01311517     +1.390488 

-  3.055297 

-0.05317053 

90 

+0.02690995      +0.00028955     +1.777154 

-  1.266575 

-0.05382187 

120 

+0.02340803     +0.01365914     +0.399475 

-  0.038680 

-0.05438924 

150 

+0.01366523     +0.02346772     -1.372927 

-  0.620648 

-0.05463405 

180 

+0.00020848     +0.02707758     -1.767867 

-  2.444894 

-0.05452574 

210 

-0.01335043     +0.02340859     -0.392395 

-  3.693094 

-0.05421519 

240 

-0.02324178     +0.01343114     +1.376698 

-  3.122646 

-0.05387086 

270 

-0.02677488      -0.00003361     +1.779939 

-  1.305659 

-0.05354950 

300 

-0.02314722      -0.01330732     +0.429867 

-  0.046772 

-0.05321706 

330 

-0.01342562      -0.02295918     -1.315326 

-  0.583802 

-0.05287735 

Zi 

+0.00034177      +0.00123615     +0.118427 

-11.073532 

-0.32182904 

22 

+0.00034182     +0.00123620     +0.118465 

-11.073535 

-0.32182921 

sin  v  •  Mi(t)  +  cos  v  •  B0<c)  =  +  0.000000000000020. 

OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS.  119 


DIFFERENTIAL  COEFFICIENTS. 

n  log  coeff. 

[defdl]M    =  +     0.12000343m'  p  9.0791936 

[dx/dt]M    =  +  63.424159      TO'  p  1.8022547 

[di/<ft]oo     =  +     0.04158807  TO'  p  8.6189687 

=  --  65.693091      TO'  n  1.8175197 

=  +  63.308999      TO'  p  1.8014655 

[dL/dt]m  =  -113.109825      TO'  n  2.0535003 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO' 

[dejdt]m    =  +o'.0000052633084 
[dx/dt}^    =  +0.0027817616 
[di/dtloo     =  +0.000001824038 

=  -0.0028812762 

=  +0.0027767109 

=  -0.0049609570 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

[de/dt]w  +0^00000  +o!o0001  +o!o00005263 

e[dirldt]M  +0.00002  +0.00002  +0.000019001 

[dildt]m  +0.00000  +0.00000  +0.000001824 

sin  i  \daldi\m  -0.000165  -0.00017  -0.000170530 


NOTES. 

That  a  division  into  eight  parts  is  here  fully  sufficient  is  shown  by  the  agreement 
of  the  final  sums.  Thus  the  greatest  effect  produced  by  all  terms  from  the  4th  to  the 
7th  order  is  seen  to  occur  with  the  coefficient  [dx/dt]0o  and  to  amount  to  but 
0".00000004. 


120 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  NEPTUNE  ON  VENUS. 


E 

A              B  cos  f           B  sin  «             0 

h 

0° 

904.77843877      +  9.109428      +21.528911      30.253556 

904.17419 

45 

904.51260261      -  6.657698      +16.194152      17.117826 

904.17356 

90 

904.39237664      -14.030230      +  1.286186      0.107979 

904.17298 

135 

904.48820486      -  8.689452      -14.462107      13.651972 

904.17345 

180 

904.74393521      +  6.236084      -21.825589      31.093114 

904.17407 

225 

905.00974680      +22.003203      -16.490829      17.750770 

904.17359 

270 

905.12994825      +29.375736       -  1.582864      0.163538 

904.17281 

315 

905.03414463      +24.034972      +14.165428      13.097596 

904.17334 

2i 

3619.04469887*     +30.691018f      -  0.593356f     61.618187 

3616.69405 

2, 

3619.04469890      +30.691025      -  0.593356      61.618164 

3616.69394 

E 

1             G             G'            G" 

e 

O      /       // 

0° 

0.53898       904.17415      0.5952295      0.0562134 

1  32  17.018 

45 

0.27377       904.17354      0.3309885      0.0571984 

1  11  14.024 

90 

0.15413       904.17298      0.1548950      0.0007710 

0  45   6.502 

135 

0.24949       904.17343      0.2998555      0.0503537 

1   7  39.561 

180 

0.50459       904.17403      0.5654480      0.0608162 

1  30  28.921 

225 

0.77088       904.17357      0.7955805      0.0246764 

1  43  33.466 

270 

0.89187       904.17281      0.8920680      0.0002028 

1  48   0.658 

315 

0.79553       904.17332      0.8133615      0.0178097 

1  44  14.706 

2, 

2.08957      3616.69397      2.2076405      0.1180034 

5  35  53.099 

22 

2.08967      3616.69386      2.2397860      0.1500382 

5  46  41.757 

ACTION  OF  NEPTUNE  ON  VENUS. 

E 

log  K0          log  La'          log  No          log  N          log  P 

logQ 

0° 

0.00023476    0.27331427    0.17644338    5.1378654    9.4986215 

2.3580297 

45 

0.00013986    0.27318776    0.17630105    5.1395214    9.5001506 

2.3595431 

90 

0.00005608    0.27307605    0.17617538    5.1436917    9.5042639 

2.3636152 

135 

0.00012618    0.27316951     0.17628052    5.1479186    9.5085363 

2.3679232 

180 

0.00022568    0.27330217    0.17642976    5.1497409    9.5104807 

2.3698895 

225 

0.00029563    0.27339543    0.17653468    5.1481063    9.5089743 

2.3683773 

270 

0.00032162    0.27343007    0.17657365    5.1439578    9.5048849 

2.3642800 

315 

0.00029957    0.27340068    0.17654059    5.1397097    9.5005900 

2.3599901 

Si 

0.00083814     1.09312256    0.70562217    0.5752556    8.0182509 

9.4558143 

22 

0.00086124     1.09315338    0.70565684    0.5752558    8.0182511 

9.4558336 

*  4o2  +  2aV  +  4[a'J  -  Zkaa'ee'  cos  K]  =  3619.04469884. 
f  4[a'V  -  kaa'e  cos  A']  =  +  30.691024. 
t  -  4k'aa'  cos  *'  •  e  sin  A''  =  -  0.593359. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


121 


ACTION  OF  NEPTUNE  ON  VENUS. 


E 

logF 

Jl'                        J2 

J> 

Ft 

0° 

2.3579959 

902.1174996      +0.0278169 

-43.343250 

-165.19752 

45 

2.3595087 

903.0233587      +1.1892143 

-32.830864 

-124.26237 

90 

2.3636147 

904.1640829      +0.1518779 

-  2.911684 

9.86927 

135 

2.3678929 

903.3068137      -1.1744691 

+28.888013 

+  110.97189 

180 

2.3698530 

902.1221024      -0.3409776 

+43.940408 

+  167.47400 

225 

2.3683624 

902.9705135      +0.9081699 

+33.428015 

+  126.53886 

270 

2.3642798 

904.1592634      +0.1562063 

+  3.508843 

+  12.14577 

315 

2.3599794 

903.2945923      -0.9279474 

-28.290864 

-108.69539 

Si 

9.4557434 

3612.5629483*     -0.0050765 

+  1.194317 

+  4.55298 

2s 

9.4557434 

3612.5952782      -0.0050323 

+  1.194300 

+  4.55299 

E 

Pi 

100000  X  Bo   100000000  X  So   1000000  X  Wo 

100000  X  B<»> 

1000000  X  «S<»> 

0° 

+0.5839042 

0.6821434    -0.4573134    -0.9883466 

0.0000000 

-0.006365877 

45 

-3.9412736 

0.6867470    +2.3281642    -0.7513860 

+0.6746065 

+0.032343148 

90 

-0.4777116 

0.6961065    +0.3193229    -0.0672756 

+0.9623607 

+0.004414608 

135 

+4.0782664 

0.7008255    -2.3820088    +0.6740551 

+0.6818058 

-0.032772474 

180 

+0.6001079 

0.7010447    -0.2565237    +1.0297346 

0.0000000 

-0.003522313 

225 

-4.0073139 

0.7005531    +2.5294517    +0.7805505 

-0.6815408 

+0.034801040 

270 

-0.5873116 

0.6967370    +0.4002346    +0.0811602 

-0.9632323 

+0.005533205 

315 

+3.9893101 

0.6879052    -2.4698966    -0.6479491 

-0.6757442 

-0.034312118 

Si 

+0.1189889 

2.7760316    +0.0057204    +0.0552726 

-0.0008716 

+0.000059623 

•y 

+0.1189890 

2.7760308    +0.0057105    +0.0552705 

-0.0008727 

+0.000059596 

E 

inonm  v  r»  „;„  ,   100000  X  -Bo  cos  w 

ItAAJlH)  X  l/VO  Sin  V                                    innnnn  ^,  lir 

„,„  ,   .            _,  100000  X  Wo  cos  u 
+  (cos  v+cos£)S0]  ,  IT   2    N  .   ™1 

\asec  *"  rl  "  °J 

100000  X  W0  sin  u 

1000  X  -  2  -  Bo 
a 

0° 

-0.00091463 

-0.68214344     -0.057911307 

-0.080091019 

-0.013549511 

45 

+  0.49123773 

-0.47994176     +0.012282960 

-0.074127847 

-0.013668483 

90 

+0.69608806 

+0.00540218     +0.005478547 

-0.003904555 

-0.013922131 

135 

+0.49653747 

+0.49458425     -0.066499046 

-0.011017258 

-0.014084333 

180 

+  0.00051305 

+0.70104468     -0.060336403 

-0.083444904 

-0.014116841 

225 

-0.49655471 

+0.49418261     +0.012013897 

-0.077124930 

-0.014078857 

270 

-0.69672347 

+0.00396738     +0.006544151 

-0.004800407 

-0.013934740 

315 

-0.49226067 

-0.48055578    -0.064023059 

-0.009971443 

-0.013691533 

Si 

-0.00103699 

+0.02827080     -0.106225012 

-0.172240885 

-0.055523223 

S2 

-0.00104018 

+0.02826932     -0.106225248 

-0.172241478 

-0.055523206 

sin  <f  •  ^A\M  +  cos 

v  .  BOM  =  -  0.0000000000000085. 

*2,W,'-G"1  =3i 

312.4449449. 

i'  -  G")  =  3612.4452400. 


122  THE  SECULAR  VARIATIONS  OF  THE  ELEMENTS 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[de/dt]w  =  -  -  0.0054696734  m'  n  7.7379614 
[dx!dt]m  =  +21.756678  TO'  p  1.3375926 
[di/dt]m  =  --  0.55945727  TO'  n  9.7477669 
[dB/(ft]oo  =-15.327159  w'  n  1.1854617 
[drldt]m  =  +21.729810  TO'  p  1.3370559 
[dL/dt]M  =  -29.268164  TO'  n  1.4663954 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO' 

[defdt]m  =  -0/.00000027764841 

[dx/dt]^  =  +0.0011044000 
[difdt]M         -0.000028398849 

[dQ/dt]w  =  -0.00077802855 

[dT/«ft]oo  =  +0.0011030360 

[dL/dt]w  =  -0.0014856935 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 
//                                  //  // 

oo    -0.00000  -0.00000  -0.00000028 

e[drldt]m    +0.00001  +0.00001  +0.00000755 

[di/dt]w     -0.00004  -0.00003  -0.00002840 

sin  ?:  [dQ/dt]M    -0.00006  -0.00005  -0.00004605 


NOTES. 

The  large  disagreement  of  the  sums  of  the  functions  near  the  beginning  of  the 
computation  is  caused,  as  in  previous  cases,  by  the  presence  of  the  term  a'V.  The 
greatest  disagreement  in  the  final  sums  occurs  in  the  second  column  and  shows  that 
the  effect  of  all  terms  from  the  4th  to  the  7th  orders  is  to  produce  a  change  of 
0".00000003  in  the  value  of  [dx/dt]00. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


123 


EARTH. 
ACTION  OF  MERCUKY  ON  THE  EARTH. 


E 

A 

B  COS  e 

It  sin  e 

1000  xg 

h 

0° 

1.25770017 

+0.37398164 

+0.15686177 

0.15586327 

1.11042819 

30 

1.20879374 

+0.24429166 

+0.30606106 

0.59337027 

1.06575342 

60 

1.14343370 

+0.05583076 

+0.37253457 

0.87910918 

1.00282212 

90 

1.07923573 

-0.14090326 

+0.33847094 

0.72569217 

0.93669980 

120 

1.03345314 

-0.29319554 

+0.21299723 

0.28738066 

0.88633014 

150 

1.01830180 

-0.36023958 

+0.02973421 

0.00560044 

0.86904450 

180 

1.03773900 

-0.32407097 

-0.16221302 

0.16667904 

0.89145124 

210 

1.08650443 

-0.19438096 

-0.31141230 

0.61430099 

0.94482277 

240 

1.15158355 

-0.00592005 

-0.37788591 

0.90454667 

1.01121409 

270 

1.21564099 

+0.19081399 

-0.34382220 

0.74882000 

1.07228810 

300 

1.26156411 

+0.34310630 

-0.21834847 

0.30200208 

1.11392105 

330 

1.27699637 

+0.41015028 

-0.03508545 

0.00779765 

1.12755062 

2, 

6.88547367* 

+0.14973214f 

-0.01605383J 

2.69558090 

6.01616683 

22 

6.88547306 

+0.14973213 

-0.01605374 

2.69558152 

6.01615921 

E 


G 


G' 


G" 


0° 

0.14093752 

1.11028337 

0.14207046 

0.00098811 

21 

1 

34.45 

30 

0.13670586 

1.06515341 

0.14124977 

0.00394390 

21 

37 

28.51 

60 

0.13427712 

1.00181061 

0.14149060 

0.00620197 

22 

30 

21.00 

90 

0.13620147 

0.93572981 

0.14260963 

0.00543817 

23 

21 

59.90 

120 

0.14078854 

0.88589477 

0.14348474 

0.00226084 

23 

53 

48.72 

150 

0.14292284 

0.86903562 

0.14297679 

0.00004507 

23 

55 

58.88 

180 

0.13995330 

0.89120229 

0.14152378 

0.00132152 

23 

34 

54.52 

210 

0.13534720 

0.94401808 

0.14077439 

0.00462250 

23 

2 

51.29 

240 

0.13403500 

1.01019211 

0.14138996 

0.00633298 

22 

24 

31.27 

270 

0.13701843 

1.07154031 

0.14266461 

0.00489838 

21 

43 

51.55 

300 

0.14130860 

1.11364215 

0.14347758 

0.00189008 

21 

9 

39.20 

330 

0.14311129 

1.12754360 

0.14316663 

0.00004830 

20 

52 

41.84 

Si 

0.83130008 

6.01302530 

0.85343712 

0.01899550 

134 

34 

49.16 

22 

0.83130709 

6.01302083 

0.85344182 

0.01899632 

134 

34 

51.97 

*  6a2  +  3aV  +  6[a'2  -  2kaa'ee'  cos  K]  =  6.8854738. 
t  6[a'V  -  kaa'e  cos  K]  =  +  0.14973211. 
|  -  Gfc'aa'  cos  <p'-e  sin  K'  =  -  0.01605375, 


124 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  MEKCUKY  ON  THE 

EARTH. 

E 

logtfo 

log  LQ 

logtfo 

log  N        log  P 

logQ 

0° 

0 

.04527343 

0.33291492 

0 

.24335185 

9. 

9618524    0.2031270 

0.1593841 

30 

0.04797642 

0.33646399 

0 

.24732687 

9. 

9917424     0.2701720 

0.2100521 

60 

0 

.05211516 

0.34189221 

0 

.25340453 

0. 

0396020     0.3745624 

0.2895406 

90 

0.05634203 

0.34742859 

0.25960067 

0.0958412    0.4959354 

0.3817747 

120 

0 

.05903754 

0.35095528 

0 

.26354624 

0. 

1435571    0.5975343 

0.4586143 

150 

0 

.05922392 

0.35119901 

0 

.26381890 

0.1631586    0.6362374 

0.4879174 

180 

0 

.05742739 

0.34884899 

0 

.26118992 

0. 

1459441    0.5935535 

0.4565142 

210 

0.05475392 

0.34534936 

0 

.25727397 

0. 

1016262     0.4927722 

0.3817985 

240 

0 

.05164961 

0.34128199 

0.25272142 

0.0482257    0.3752715 

0.2938290 

270 

0 

.04846610 

0.33710662 

0.24804652 

0. 

0004822    0.2736102 

0.2165394 

300 

0.0-1587421 

0.33370403 

0 

.24423576 

9.9673367    0.2060765 

0.1640904 

330 

0 

.04461832 

0.33205428 

0.24238775 

9. 

9536821     0.1814324 

0.1439179 

2l 

0 

.31137734 

2.04959742 

1 

.51844973 

0.3065179    2.3501251 

1.8219725 

Sj 

0 

.31138070 

2.04960185 

1 

.51845468 

0. 

3065326    2.3501595 

1.8219999 

E 

logF 

Ji' 

J, 

J, 

F, 

0° 

0.1589092 

0.14310992 

+0.032517759 

+0.009293505 

-0.004694216 

30 

0.2080805 

0.14534318 

+0.064629382 

+0.012521451 

-0.009159128 

60 

0.2862522 

0.14795099 

+0.079427870 

+0.011088166 

-0.011148401 

90 

0.3786919 

0.14823938 

+0.072184328 

+0.005377209 

-0.010129019 

120 

0.4572602 

0.14575314 

+0.044711897 

+0.003081410 

-0.006374115 

150 

0.4878898 

0.14319523 

+0.004967377 

-0.012020967 

-0.000889820 

180 

0.4557266 

0.14344333 

-0.035694279 

-0.019045640 

+0.004854357 

210 

0.3791987 

0.14602945 

-0.066230415 

-0.022272938 

+0.009319268 

240 

0.2904985 

0.14803624 

-0.078997000 

-0.020838344 

+0.011308544 

270 

0.2141065 

0.14763142 

-0.071296738 

-0.015126732 

+0.010289159 

300 

0.1631852 

0.14536909 

-0.045398479 

-0.006668768 

+0.006534256 

330 

0.1438950 

0.14322488 

-0.007686344 

+0.002269483 

+0.001049961 

Si 

1.8118319 

0.87366271* 

-0.003432232 

-0.029252491 

+0.000480425 

2i 

1.8118624 

0.87366354 

-0.003432410 

-0.029252494 

+0.000480421 

*s,(J 

V  -  G")  = 

0.85466721. 

St(Ji  -  G")  = 

0.85466722. 

OF   THE    ORBITS   OF   THE   FOUR   INNER   PLANETS. 


125 


ACTION  OF  MERCURY  ON  THE  EARTH. 

E 

1000  X  F3        Ra           So            Wo           «<"> 

gto) 

0° 

-0.3347370   -0.9146346   +0.03939096   +0.012865148    0.0000000 

+0.04006286 

30 

-0.1102483   -0.9755942   +0.08729250   +0.020012484   -0.4949864 

+0.08857904 

60 

+0.5661785   -1.0850638   +0.12713088   +0.022775691   -0.9476313 

+0.12820596 

90 

+  1.0162060   -1.2358930   +0.14090497   +0.016043819   -1.2358930 

+0.14090497 

120 

+0.7859945   -1.3865198   +0.10290744   -0.005719611   -1.1907761 

+0.10205179 

150 

+0.1010653   -1.4553467   +0.01142556   -0.036530916   -0.7172557 

+0.01126198 

180 

-0.3579653   -1.3946588   -0.08289463   -0.055794474    0.0000000 

-0.08152732 

210 

-0.1348464   -1.2530893   -0.12959913   -0.053749951   +0.6175749 

-0.12774376 

240 

+0.5468018   -1.1065943   -0.12737525   -0.039380718   +0.9503695 

-0.12631602 

270 

+  1.0072426   -0.9942912   -0.09740907   -0.022874526   +0.9942912 

-0.09740907 

300 

+0.7898462   -0.9252302   -0.05560180   -0.008440800   +0.8080489 

-0.05607200 

330 

+0.1166998   -0.8987688   -0.00911126   +0.003338197   +0.4560076 

-0.00924555 

Zi 

+  1.9961187   -6.8127015   +0.00355760   -0.073694764   -0.3799890 

+0.00640527 

* 

+  1.9961190   -6.8129832   +0.00350357   -0.073760893   -0.3802614 

+0.00634761 

E 

„  .           —  Rocosv 

Ho  sm  v                            ,,,    ,    .       ,„  .  .    . 
m  o   ,  lr   i   \  i\  •   o     "  o  cos  (v  +  JT)       TV  o  sm  (v  +  IT) 
+  (cosv+cosE)S0  +(-sec2  ip  +  llsmv<So 

-2-fl 
o 

0° 

+0.0787819     +0.9146346     -0.002313800     +0.012655370 

1.7985901 

30 

-0.3440930     +0.9286742     -0.013088473     +0.015139066 

1.9228487 

60 

-0.8219877     +0.7498962     -0.021560174     +0.007341046 

2.1519297 

90 

-1.2380823     +0.2610826     -0.015731601     -0.003149764 

2.4717860 

120 

-1.2947997     -0.5330552     +0.004304192     +0.003766681 

2.7962929 

150 

-0.7369917     -1.2550382     +0.011990880     +0.034506913 

2.9529690 

180 

+0.1657893     -1.3946588     -0.010034649     +0.054884684 

2.8360980 

210 

+0.8424959     -0.9617143     -0.034466377     +0.041244714 

2.5425788 

240 

+  1.0791999     -0.3473975     -0.036894585     +0.013770630 

2.2317474 

270 

+0.9957850     +0.1781428     -0.022567368     -0.003736033 

1.9885826 

300 

+  0.7530386     +0.5475911     -0.006510755     -0.005371886 

1.8349430 

330 

+0.4402010     +0.7837111     +0.001148460     +0.003134422 

1.7714297 

Z, 

-0.0399777     -0.0629896     -0.073009771     +0.087046525 

13.6496011 

2j 

-0.0406851     -0.0651418     -0.072714479     +0.087139318 

13.6501948 

sin  <p  •  J.Ai('>  +  cos  ?  •  BoM  =  +  0.00000006. 

126  THE  SECULAR  VARIATIONS  OP  THE  ELEMENTS 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[de!dt]w  =  -        8710J780TO'  n  3.9400270 

[dx/dt]M  =  [dTT/dt]m  =--824986.23      TO'  n  5.9164467 

[dp/dt]*,  =  +     18814.333    TO'  p  4.2744888 

[dqfdt]m  =  -       15740.112    TO'  n  4.1970078 

[dL/dt]w  =  +2948201.7       w'  p  6.4695572 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 


=  -o(X)l  1613570 

[dx/dt]m  =  [drfdt]m    =  -0.10999815 
[dpfdt]M    =  +0.0025085775 
[dq/dt]w    =  -0.0020986812 
[dL/dt]M  =  +0.39309355 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


[de/dt]m 

-0.00116 

-0.00116 

-0.001  16136 

e[dir/dt]w 

-0.00184 

-0.00184 

-0.00184479 

[dp/dt]M 

+0.00250 

+0.00251 

+0.00250858 

[dq/dt]w 

-0.00209 

-0.00210 

-0.00209868 

[dL/dt]w 

+0.3931 

+0.39309355 

NOTES. 

Although  /  and  e'  are  here  very  large,  the  error  in  the  approximate  test  with 
e,  G,  G',  G"  and  6  is  small  in  consequence  of  the  smallness  of  the  factor  a'.  As  we 
approach  the  end  of  the  computation,  however,  the  difference  of  the  sums  steadily 
increases,  indicating  the  rather  slow  convergence  of  the  perturbing  function.  The 
greatest  difference  is  in  the  coefficient  [dTr/dt]OQ  where  terms  from  the  fifth  to  the 
eleventh  orders  inclusive  amount  to  one  sixtieth  part  of  the  remaining  terms  and 
produce  an  effect  of  0".0018  in  the  value  of  [dir/dt]00.  A  division  into  twelve  parts 
is  thus  necessary  in  this  case,  but  a  comparison  with  the  computation  of  the  action  of 
Mars  on  Mercury,  and  especially  with  the  similar  case  of  Mercury  on  Venus,  where 
twenty-four  points  of  division  are  employed,  renders  it  evident  that  more  than  twelve 
points  are  in  the  present  case  unnecessary. 


OF   THE    ORBITS    OF   THE    FOUR   INNER   PLANETS. 


127 


The  agreement  with  previous  values  is  exact.     The  results  obtained  by  HILL 
in  the  "New  Theory,11  pages  511  and  512,  are, 


These  are,  however, 
Venus  on  the  Earth. 


=  +0.0025049     [dq/dt]m  =  -0.0020956 
but  provisional -values.     (See  the  note  to  the  computation  of 


E 

A 

ACTION 
logB 

OF  VENUS  ON 
i 

THE  EARTH. 

e' 

0 

o    / 

,, 

O 

1 

H 

0 

1.49844749 

9.8537612 

331   0 

3.89      0.07950559 

7 

0 

55.62 

30 

1.50411382 

9.8546442 

1  19 

46.170     0.08070333 

5 

48 

28.79 

60 

1.51484369 

9.8567645 

31  34 

37.20      0.08262793 

4 

19 

13.62 

90 

1.52786583 

9.8597314 

61  38 

57.20      0.08465948 

3 

12 

16.55 

120 

1.53974201 

9.8626537 

91  29 

35.875     0.08634022 

2 

33 

43.11 

150 

1.54723847 

9.8644870 

121   8 

43.24      0.08741009 

2 

9 

29.44 

180 

1.54824407 

9.8645724 

150  43 

29.14      0.08768573 

1 

52 

5.43 

210 

1.54243719 

9.8629683 

180  22 

35.176     0.08700647 

1 

54 

16.46 

240 

1.53142595 

9.8603568 

210  12 

5.09      0.08536569 

2 

37 

52.10 

270  ' 

1.51826327 

9.8576205 

240  13 

17.66      0.08309965 

4 

8 

28.42 

300 

1.50652764 

9.8554195 

270  23 

50.222     0.08090038 

5 

56 

51.93 

330 

1.49931255 

9.8540767 

300  40 

21.95      0.07954648 

7 

7 

42.75 

Z, 

9.13923084* 

9.1535280 

1085  23 

41.42      0.50242554 

24 

20 

41.81 

22 

9.13923113 

9.1535281 

905  23 

41.40      0.50242550 

24 

20 

42.41 

E 

1000  X  r' 

lOOOXs 

G 

G'      1000000  X  G" 

e 

O 

O 

, 

,, 

0 

2.7380635 

0.002936418 

0.97594422 

0.522484505     5.79 

47 

1 

40.84 

30 

2.3200483 

0.000006753 

0.98353171 

0.520557558     0.04 

46 

40 

41.61 

60 

1.7893068 

0.003473631 

0.99543723 

0.519388564     6.67 

46 

14 

52.33 

90 

1.3770128 

0.009946420 

1.00773136 

0.520128988    18.98 

45 

55 

31.27 

120 

1.1340363 

0.013007536 

1.01774388 

0.521998108    24.54 

45 

44 

23.02 

150 

0.9732049 

0.009615238 

1.02406979 

0.523162060    17.95 

45 

37 

23.13 

180 

0.8464697 

0.003140184 

1.02571566 

0.522509836     5.92 

45 

32 

21.65 

210 

0.8529468 

0.000000563 

1.02173849 

0.520674148     0.00 

45 

32 

59.47 

240 

1.1449685 

0.003259143 

1.01199593 

0.519411649     6.09 

45 

45 

34.89 

270 

1.7299249 

0.009581104 

0.99828868 

0.519968638    18.53 

46 

11 

45.77 

300 

2.3843853 

0.012589278 

0.98467983 

0.521847851    24.49 

46 

43 

7.27 

330 

2.7841227 

0.009256115 

0.97615654 

0.523149586    18.07 

47 

3 

38.83 

Zi 

10.0372301 

0.038406190 

6.01151675 

3.127640513    73.50 

277 

2 

0.00 

22 

10.0372604 

0.038406192 

6.01151657 

3.127640978    73.57 

277 

2 

0.08 

*  6o2  +  3a2e2  +  6[o'2  - 

2kaa'ce'  cos  K]  =  9 

13923110. 

128 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  VENUS  ON  THE  EARTH. 

E 

logtfo 

log  Lo' 

log  AT0          log  N          log  P 

logO 

0° 

0.26147483 

0.60779696 

0.54775634    0.2626427    0.8915846 

0.8209715 

30 

0.25683878 

0.60207812 

0.54149917    0.2549482    0.8714495 

0.8036590 

60 

0.25122074 

0.59513848 

0.53390180    0.2468811     0.8459860 

0.7827661 

90 

0.24707162 

0.59000660 

0.52828039    0.2420422    0.8253429 

0.7669697 

120 

0.24470700 

0.58707937 

0.52507272    0.2404872    0.8122693 

0.7579113 

150 

0.24322990 

0.58524987 

0.52306751     0.2402489    0.8048244 

0.7529792 

180 

0.24217345 

0.58394092 

0.52163264    0.2400756    0.8019574 

0.7506787 

210 

0.24230579 

0.58410492 

0.52181242    0.2408209    0.8062462 

0.7532935 

240 

0.24496050 

0.58739327 

0.52541673    0.2444416    0.8214721 

0.7646769 

270 

0.25055055 

0.59430993 

0.53299440    0.2516543    0.8474359 

0.7853846 

300 

0.25737182 

0.60273603 

0.54221917    0.2600985    0.8762226 

0.8090118 

330 

0.26191246 

0.60833646 

0.54834645    0.2649133    0.8941948 

0.8237322 

2i 

1.50190834 

3.56408503 

3.19599940    1.4946267    5.0494920 

4.6860161 

S2 

1.50190910 

3.56408590 

3.19600034     1.4946278    5.0494937 

4.6860181 

E 

log  V 

Ji' 

1000  X  Ji         1000  X  J3 

1000  X  Ft 

0° 

0.8209685 

0.522862677 

-  3.0632731      +  1  2.629050 

+  1.2372987 

30 

0.8036589 

0.521939851 

-0.7336504      +24.877829 

-0.0593369 

60 

0.7827628 

0.521368465 

+2.4504401      +30.338229 

-1.3457284 

90 

0.7669603 

0.521723193 

+5.0835797      +27.547130 

-2.2771885 

120 

0.7578992 

0.522632205 

+5.8345100      +17.252379 

-2.6041325 

150 

0.7529704 

0.523191642 

+4.4234178      +  2.212471 

-2.2389562 

180 

0.7506758 

0.522862807 

+  1.7386808      -13.542647 

-1.2795084 

210 

0.7532935 

0.521968683 

-0.9069972      -25.791387 

+0.017127o 

240 

0.7646739 

0.521377068 

-2.6859269      -31.251723 

+  1.3035189 

270 

0.7853753 

0.521675619 

-3.5987100      -28.460585 

+2.2349789 

300 

0.8089994 

0.522582150 

-4.0331250      -18.165873 

+  2.5619225 

330 

0.8237230 

0.523186452 

-4.0263352       -  3.126030 

+  2.1967465 

2, 

4.6859794 

3.133685372* 

+0.2413060       -  2.740585 

-0.1266292 

22 

4.6859813 

3.133685439 

+0.2413047       -  2.740573 

-0.1266288 

*  S,(J,'  -  G")  =  3. 

133611872. 

2,(J,'  -  G")  =  3. 

133611869. 

OF   THE    ORBITS   OF   THE   FOUR   INNER   PLANETS. 


129 


ACTION  OF  VENUS  ON  THE  EARTH. 

E 

1000  XF3 

RO      100  x  So      w,,       --RO 

«<»> 

a 

0° 

+0.06535471 

-  1.8283274   -1.0644424   +0.08413477    1.7976643 

0.0000000 

30 

-0.00196302 

-  1.7898617   -0.5109526   +0.15828193    1.7638656 

-0.9081206 

60 

-0.00570449 

-  1.7535729   +0.5420211   +0.18393250    1.7388680 

-1.5314810 

90 

+0.05916590 

-  1.7367356   +1.4494077   +0.16147390    1.7367356 

-1.7367356 

120 

+0.12885901 

-  1.7362036   +1.6510300   +0.09963433    1.7507629 

-1.4910928 

150 

+0.13426042 

-  1.7386892   +1.0760758   +0.01338361    1.7639423 

-0.8568990 

180 

+0.06988985 

-  1.7361351   +0.1682926   -0.07583153    1.7652520 

0.0000000 

210 

-0.00059748 

-  1.7337538   -0.5029615   -0.14614352    1.7589352 

+0.8544665 

240 

-0.00787443 

-  1.7442502   -0.6981624   -0.18183297    1.7588768 

+  1.4980034 

270 

+0.05404190 

-  1.7747204   -0.6225129   -0.17324740    1.7747204 

+  1.7747204 

300 

+0.12215392 

-  1.8154792   -0.6714245   -0.11610019    1.8002552 

+  1.5855471 

330 

+0.12777090 

-  1.8402326   -0.9613070   -0.01982985    1.8135046 

+0.9336770 

?i 

+0.37267857 

-10.6139684   -0.0726856   -0.00606309   10.6116792 

+0.0609767 

22 

+0.37267862 

-10.6139933   -0.0722505   -0.00608133   10.6117037 

+0.0611087 

,-,   .                —  Ra'cOS  V 

M 

S« 

+  (cost)+cosE)So  +  (  -sec!^+lJsint>/So 

WL,  sin  (»  +  IT) 

0° 

-0.010825990 

-0.0212889     +1.8283274     -0.01513166 

+0.08276286 

30 

-0.005184831 

-0.9168211     +1.5373039     -0.10351881 

+0.11973727 

60 

+0.005466048 

-1.5259137     +0.8639706     -0.17411620 

+0.05928500 

90 

+0.014494077 

-1.7367344     -0.0001389     -0.15833153 

-0.03170097 

120 

+0.016373004 

-1.5075995     -0.8612810     -0.07497804 

-0.06561473 

150 

+0.010606704 

-0.8754611     -1.5022509     -0.00439303 

-0.01264209 

180 

+0.001655167 

-0.0033659     -1.7361351     -0.01363832 

+0.07459502 

210 

-0.004957609 

+0.8630786     -1.5036464     -0.09371241 

+0.11214236 

240 

-0.006923567 

+  1.5048615     -0.8818403     -0.17035373 

+0.06358327 

270 

-0.006225129 

+  1.7745751     -0.0173138     -0.17092102 

-0.02829602 

300 

-0.006771025 

+  1.5786949     +0.8963895     -0.08955312 

-0.07388837 

330 

-0.009754749 

+0.9169364     +1.5955426     -0.00682218 

-0.01861936 

2i 

-0.001026363 

+0.0253885     +0.1094311     -0.53777107 

+0.14072305 

2i 

-0.001021537 

+0.0255735     +0.1094965     -0.53769898 

+0.14062119 

sin  <p  •  \A\W  +  cos  (f 

•  Bo(c>  =  -  0.0000000083. 

130  THE  SECULAR  VARIATIONS  OF  THE  ELEMENTS 

DIFFERENTIAL  COEFFICIENTS. 

u  log  coeff. 

[de/dt]w    =  +      5503.0089  m'  p  3.7406002 

[dx/dt\  =  [dirfdt]      =  +1409586.4       TO'  p  6.1490917 

[dp/dt}^    =  +     30388.832    TO'  p  4.4827140 

[dq/dt]m    =  -     116164.73      TO'  n  5.0650743 

=  +4584354.6       TO'  p  6.6612782 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF 

[de/(ft]M  =  +  0.013483339 

[dx/dt]  =  [dTr/dt]w  =  +  3.4537341 

[dp/dt]w  =  +  0.074457966 

[dq/dt]oo  =   •-  0.28462399 

[dL/dt]w  =  +11.232473 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


TO' 


[de/dt]m 

+  0.01344 

+o'.01348 

+  0.0134833 

e[dw/dt] 

+  0.05796 

+0.05792 

+  0.0579231 

[dpldt]M 

+  0.07450 

+0.07446 

+  0.0744580 

[dq/dt]oo 

-  0.28454 

-0.28462 

-  0.2846240 

[dL/dt]w 

+  11.2298 

+  11.232473 

NOTES. 

This  computation  is  of  special  interest  because,  notwithstanding  the  low  eccen- 
tricities of  both  the  Earth  and  Venus,  the  perturbing  function  is  but  slowly  con- 
vergent for  this  case.  In  1893,  the  computation  was  effected  by  MR.  R.  T.  A.  INNES 
who  employed  HILL'S  second  modification  of  GAUSS'S  method,  using  in  the  work 
manuscript  tables  prepared  by  himself.  (See  M.  N.,  Vol.  LIII,  No.  6.  The  tables 
were  afterward  published  in  M.  N.,  Vol.  LIV,  No.  5.)  The  values  of  [dp/dt]00  and 
[dq/dt]oo  were  also  obtained  by  HILL  in  the  "New  Theory,"  pages  511  and  512. 

As  the  results  of  INNES  differed  considerably  in  some  cases  from  those  hitherto 
obtained,  particularly  in  the  case  of  [de/dt]0o,  which  agreed  to  the  first  two  figures 
only  with  the  values  of  LEVERRIER  and  NEWCOMB,  and  in  the  case  of  [dq/dt]QO,  which 


OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  131 

differed  in  the  fourth  figure  from  the  value  given  by  HILL,  and  in  order  to  make  the 
comparison  more  exact,  the  roots  in  the  present  paper  were  computed  by  the  formulas 
of  the  second  method,  their  values  being  afterward  verified  by  those  of  the  first.  It 
was  found  that  the  functions  tabulated  by  MR.  INNES  are  substantially  correct, 
though  the  last  two  significant  figures  of  all  functions  from  R  0  to  the  end  usually 
differ,  doubtless  owing  to  the  inaccuracy  of  the  tables  employed  by  MR.  INNES. 
Using  the  values  as  given  by  him,  all  of  this  part  of  his  computation  was  duplicated, 
with  the  result  that  an  error  was  found  in  his  value  of  [de/dt]00,  while  for  [dq/dt]0a 
and  the  other  coefficients  his  values  were  found  to  be  substantially  correct.  The 
various  values  here  referred  to  are  as  follows: 

Innes.  Hill. 


[de/dt]oo 
e[dw/dt]m 
[dp/dflw, 
[dq/dtloo 

+  0.013476* 
+  0.057915 
+  0.074459 
-  0.284623 
+  11.232490 

a 
+0.0744329 
-0.2845280 

It  will  be  noticed  that  the  results  of  INNES  are  in  almost  exact  accordance  with 
those  here  given.  The  disagreement  of  the  value  of  [dqldt]00  as  derived  by  GAUSS'S 
method  with  that  found  by  HILL  is,  however,  a  more  serious  matter,  and  is  almost 
the  sole  cause  of  the  considerable  disagreement  of  the  values  of  this  variation  in  the 
complete  perturbations  of  the  Earth's  orbit,  the  values  of  [dqldt]OQ  from  the  action 
of  all  of  the  other  planets  except  Venus  agreeing  with  those  obtained  by  HILL  very 
exactly.  Using  the  values  tabulated  on  page  510  of  the  "New  Theory  "  and  the 
formulas  of  page  511,  I  have  duplicated  the  computation  by  HILL'S  methods  and 
find  the  same  results  as  he  obtained.  It  is  to  be  noticed  that  the  theory  of  the  motion 
of  the  ecliptic  here  given  by  HILL  was  to  serve  a  temporary  purpose  only,  the  numerical 
values  of  the  coefficients  stated  by  LEVERRIER  in  the  Annales,  Vol.  II,  pages  94  to  96, 
being  employed  without  a  re-computation  of  them. 

*  The  uncorrected  value  was  +  0".013156. 


132 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  MARS  ON  THE  EARTH. 


E 

A 

B  COS  e 

/;  -in  • 

g 

h 

0° 

3.12005845 

-0.6857946 

+  1.1901000 

0.028604007 

2.3106194 

22.5 

3.04885416 

-1.0809801 

+0.7480691 

0.011301685 

2.3059358 

45 

3.01959381 

-1.2762880 

+0.1890610 

0.000721878 

2.3030821 

67.5 

3.03677529 

-1.2419833 

-0.4018201 

0.003260792 

2.3034537 

90 

3.09782583 

-0.9832901 

-0.9346186 

0.017641228 

2.3070230 

112.5 

3.19346899 

-0.5395909 

-1.3282200 

0.035628705 

2.3122967 

135 

3.30912609 

+0.0215645 

-1.5227032 

0.046826348 

2.3167796 

157.5 

3.42714623' 

+0.6147456 

-1.4884581 

0.044743825 

2.3179851 

180 

3.52951891 

+  1.1496460 

-1.2306997 

0.030588915 

2.3148501 

202.5 

3.60064084 

+  1.5448315 

-0.7886687 

0.012561720 

2.3089820 

225 

3.62970226 

+  1.7401394 

-0.2296607 

0.001065206 

2.3043625 

247.5 

3.61232195 

+  1.7058351 

+0.3612206 

0.002635147 

2.3044350 

270 

3.55118902 

+  1.4471417 

+0.8940191 

0.016141855 

2.3088270 

292.5 

3.45562818 

+  1.0034423 

+  1.2876203 

0.033483868 

2.3139342 

315 

3.34017005 

+0.4422870 

+  1.4821032 

0.044362582 

2.3163153 

337.5 

3.22234876 

-0.1508942 

+  1.4478582 

0.042336214 

2.3148421 

S, 

26.59718442* 

+  1.8554059f 

-0.16239891 

0.185952019 

18.4818589 

22 

26.59718440 

+  1.8554060 

-0.1623987 

0.185951956 

18.4818645 

*  8as  +  4a8e'  +  8[a'!  -  2kaa'ee'  cos  A']  =  26.59718442. 
t  8[a'V  -  kaa'e  cos  K]  =  +  1.8554056. 
t  -  Sk'aa'  cos  v'  •  e  sin  A"  =  -  0.1623983. 


E 


OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS. 

ACTION  OF  MAES  ON  THK  EARTH. 

G  G'  G" 


133 


0° 
22.5 
45 
67.5 
90 
112.5 
135 
157.5 
180 
202.5 
225 
247.5 
270 
292.5 
315 
337.5 

2, 

?2 

0.7892434 
0.7227227 
0.6963160 
0.7131259 
0.7706071 
0.8609765 
0.9721508 
.   1.0889654 
1.1944730 
1.2714631 
1.3051440 
1.2876912 
1.2221663 
1.1214982 
1.0036591 
0.8873110 

7.9537596 
7.9537539 

2.3024091      0.8127396      0.0152860       36  42  22.96 
2.3028298      0.7325284      0.0066997       34  27  16.75 
2.3028870      0.6969608      0.0004498       33  23   4.72 
2.3025627      0.7159948      0.0019779       33  55  44.11 
2.3020189      0.7853689      0.0097577       35  54  24.65 
2.3015508      0.8891330      0.0174106      38  41  59.61 
2.3014739      1.0076482      0.0201918       41  42  38.47 
2.3019609      1.1223087      0.0173190       44  30  20.12 
2.3028661      1.2173683      0.0109112       46  46   8.30 
2.3036994      1.2810025      0.0042567       48  15  58.20 
2.3038996      1.3059610      0.0003540       48  50  43.66 
2.3033085      1.2897047      0.0008871       48  27   7.65 
2.3023363      1.2343370      0.0056800       47   8  13.64 
2.3016065      1.1465149      0.0126889       45   3   3.25 
2.3014626      1.0370980      0.0185863       42  25  11.06 
2.3018396      0.9202987      0.0199852       39  31  20.00 

18.4193535      8.0974817      0.0812168      332  52  47.46 
18.4193582      8.0974855      0.0812251      332  52  49.69 

ACTION  OF  MAKS  ON  THE  EARTH. 

E 

log  A'0 

log  L0' 

log  Na 

log  AT 

logP 

logQ 

0° 

0.14793515 

0.46562072 

0.39121913 

9.5856600 

9.3211682 

9.6118228 

22.5 

0.12867513 

0.44104101 

0.36395436 

9.5698260 

9.2838199 

9.5702568 

45 

0.12011431 

0.43007010 

0.35176641 

9.5662159 

9.2715711 

9.5556249 

67.5 

0.12442259 

0.43559482 

0.35790544 

9.5749532 

9.2853792 

9.5702743 

90 

0.14089818 

0.45665643 

0.38128232 

9.5949794 

9.3237440 

9.6123158 

112.5 

0.16647531 

0.48915019 

0.41726416 

9.6240919 

9.3826551 

9.6760625 

135 

0.19735020 

0.52805532 

0.46020914 

9.6588907 

9.4553466 

9.7533001 

157.5 

0.22941061 

0.56809820 

0.50425204 

9.6947360 

9.5321279 

9.8336349 

180 

0.25803549 

0.60355500 

0.54311538 

9.7259997 

9.6009117 

9.9047936 

202.5 

0.27840311 

0.62861983 

0.57051120 

9.7469169 

9.6490816 

9.9542006 

225 

0.28660968 

0.63868139 

0.58149048 

9.7530543 

9.6666751 

9.9720145 

247.5 

0.28101715 

0.63182707 

0.57401211 

9.7427949 

9.6495832 

9.9542876 

270 

0.26293394 

0.60959543 

0.54972342 

9.7180756 

9.6011932 

9.9045601 

292.5 

0.23607945 

0.57638319 

0.51334444 

9.683858o 

9.5314041 

9.8327841 

315 

0.20515940 

0.53784176 

0.47098807 

9.6465515 

9.4533990 

9.7520425 

337.5 

0.17455362 

0.49936282 

0.42855178 

9.6122456 

9.3799495 

9.6749680 

V 
—  1 

1.61903635 

4.27007615 

3.72979435 

7.2494269 

5.6940086 

8.0664741 

22 

1.61903696 

4.27007713 

3.72979553 

7.2494229 

5.6940004 

8.0664686 

134 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  MARS  ON  THE  EARTH. 


E 

logF 

Ji' 

J2 

J, 

Ft 

0° 

9.6084099 

2.3152378 

+0.11029083 

+0.061812671 

-0.25644059 

22.5 

9.5687497 

2.3059045 

+0.06946998 

+0.074883922 

-0.16119255 

45 

9.5555233 

2.2995328 

+0.01805514 

+0.077080482 

-0.04073852 

67.5 

9.5698283 

2.3016313 

-0.03654213 

+0.068068172 

+0.08658346 

90 

9.6101298 

2.3103260 

-0.08632858 

+0.049219187 

+0.20138993 

112.5 

9.6721978 

2.3186977 

-0.12376087 

+0.023403207 

+0.28620250 

135 

9.7488617 

2.3215901 

-0.14288822 

-0.005449606 

+0.32810925 

157.5 

9.8298619 

2.3181659 

-0.14041555 

-0.032946848 

+0.32073022 

180 

9.9024333 

2.3108630 

-0.11642269 

-0.054902532 

+0.26518896 

202.5 

9.9532839 

2.3034774 

-0.07451078 

-0.067974156 

+0.16994090 

225 

9.9719383 

2.2994233 

-0.02127963 

-0.070171605 

+0.04948689 

247.5 

9.9540965 

2.3004768 

+0.03479475 

-0.061160169 

-0.07783516 

270 

9.9033316 

2.3061702 

+0.08485465 

-0.042311544 

-0.19264158 

292.5 

9.8300222 

2.3139376 

+0.12119708 

-0.016495194 

-0.27745411 

315 

9.7479652 

2.3199983 

+0.13850882 

+0.012358484 

-0.31936096 

337.5 

9.6705453 

2.3208585 

+0.13455789 

+0.039856619 

-0.31198183 

s, 

8.0485930 

18.4831415* 

-0.01520968 

+0.027635537 

+0.03499338 

22 

8.0485855 

18.4831497 

-0.01520963 

+0.027635553 

+0.03499343 

*S,(y,'  -G")  =  18.4019247. 
Zt(Ji'  -  G")  =  18.4019246. 


OF   THE    ORBITS    OF   THE   FOUR   INNER   PLANETS. 


135 


ACTION  OF  MARS  ON  THE  EARTH. 

E 

1000  X  Ft 

flo 

-So 

w. 

RW 

gw 

0° 

-  5.019874 

0.22207195 

-0.00895625 

+0.02403761 

0.00000000 

-0.00910901 

22.5 

-  1.341522 

0.21055435 

-0.00524961 

+0.02748415 

+0.08184381 

-0.00533223 

45 

+  0.173061 

0.20752103 

-0.00112516 

+0.02773157 

+0.14850060 

-0.00113867 

67.5 

-  1.404619 

0.21281662 

+0.00313244 

+0.02500875 

+0.19788699 

+0.00315267 

90 

-  5.191525 

0.22627677 

+0.00726155 

+0.01896286 

+0.22627677 

+0.00726155 

112.5 

-  9.004074 

0.24749832 

+0.01089521 

+0.00882887 

+0.22720045 

+0.01082573 

135 

-10.631982 

0.27522212 

+0.01347761 

-0.00609013 

+  0.19233060 

+0.01331965 

157.5 

-  9.129495 

0.30647387 

+0.01430898 

-0.02537639 

+0.11549298 

+0.01409065 

180 

-  5.368216 

0.33606721 

+0.01279803 

-0.04599728 

0.00000000 

+0.01258694 

202.5 

-  1.527816 

0.35740451 

+0.00883760 

-0.06172287 

-0.13468590 

+0.00870276 

225 

+  0.177177 

0.36477534 

+0.00302208 

-0.06569865 

-0.25491209 

+0.00298666 

247.5 

•  1.210722 

0.35578971 

-0.00342959 

-0.05556595 

-0.32661056 

-0.00340772 

270 

-  4.837361 

0.33265813 

-0.00898150 

-0.03579916 

-0.33265813 

-0.00898150 

292.5 

-  8.543560 

0.30146830 

-0.01237466 

-0.01405700 

-0.28031952 

-0.01245460 

315 

-10.135232 

0.26942963 

-0.01319021 

+0.00403826 

-0.19280199 

-0.01334851 

337.5 

-  8.672134 

0.24201044 

-0.01181400 

+0.01658570 

-0.09407098 

-0.01199994 

2, 

-40.833952 

2.23402218 

+0.00430615 

-0.07881492 

-0.21326424 

+0.00357711 

2s 

-40.833942 

2.23401612 

+0.00430637 

-0.07881474 

-0.21326273 

+0.00357732 

sin 

<p  •  l-Ai(a)  +  cos 

V  •  Bo«>  =  + 

0.0000000050. 

136 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  MARS  ON  THE  EARTH. 


E 

Ro  sin  v  + 
(cos  v  +  cos  E)  So 

(  -  sec'  if  +  ij  sin  vSa 

Wo  COS  (!)+«•) 

Wo  sin  (v  +  T  ) 

-2a/J0 

0° 

-0.01791249 

-0.22207195 

-0.00432317 

+0.02364565 

-0.43669510 

22.5 

+0.07214538 

-0.19805102 

-0.01506202 

+0.02298943 

-0.41458381 

45 

+0.14689806 

-0.14657925 

-0.02300258 

+0.01548939 

-0.41012014 

67.5 

+0.20021147 

-0.07256855 

-0.02452836 

+0.00487823 

-0.42290136 

90 

+0.22612318 

+0.01831801 

-0.01859384 

-0.00372284 

-0.45255354 

112.5 

+0.21867465 

+0.11830141 

-0.00734127 

-0.00490454 

-0.49817351 

135 

+0.17313163 

+0.21584084 

+0.00340234 

+0.00505112 

-0.55697199 

157.5 

+0.08900258 

+0.29475425 

+0.00517812 

+0.02484247 

-0.62244500 

180 

-0.02559607 

+0.33606721 

-0.00827262 

+0.04524724 

-0.68340683 

202.5 

-0.15101812 

+0.32435071 

-0.03316015 

+0.05205879 

-0.72588458 

225 

-0.25917515 

+0.25670929 

-0.05360955 

+0.03797800 

-0.73820246 

247.5 

-0.32389097 

+0.14753236 

-0.05413656 

+0.01252224 

-0.71614631 

270 

-0.33246073 

+0.02354205 

-0.03531845 

-0.00584698 

-0.66531623 

292.5 

-0.28957300 

-0.08808428 

-0.01192486 

-0.00744291 

-0.59906678 

315 

-0.21131668 

-0.16946344 

+0.00233483 

+0.00329486 

-0.53246901 

337.5 

-0.11585773 

-0.21387153 

+0.00359251 

+0.01619196 

-0.47652121 

s, 

-0.30030825 

+0.31236276 

-0.13738304 

+0.12113644 

-4.47573530 

22 

-0.30030574 

+0.31236335 

-0.13738259 

+0.12113567 

-4.47572256 

DIFFERENTIAL  COEFFICIENTS. 


[dxldt]m  = 


=  -     48641.893m' 
=  +3016769.1      m' 


log  coeff. 
n  4.6870105 
p  6.4795421 


=  + 
[dq/dt]w    = 

=  -  724628.93 


19626.398  m'     p  4.2928406 
22258.695  m'    n  4.3474997 


m'    n  5.8601 157 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  m'. 


=  -0.015723904 
[dx/dt]m  =  [d7r/d<]oo    =  +0.97519611 

[dpfdt]M  =  +0.0063443986 
[dq/dt]m  =  -0.0071953108 
[dL/dt]m  =  -0.23424243 


OF   THE    ORBITS    OF   THE    FOUR   INNER   PLANETS. 


137 


Leverrier. 

•[<fe/cft]oo  -0.01573 

[dw/dt]m  +0.9754 

[dp/dt]w  +0.00635 

[dq/dtlw  -0.00721 

[dL/dt]w  -0.2337 


COMPARISON  WITH  OTHER  RESULTS. 

Innes.  Hall.                    Newcomb.  Method  of  Gauss. 

-0.015722  -0.0157232  -0.01572  -0.0157239 

+0.975224  +0.9751387  +0.9755  +0.9751961 

+0.0063401  +0.0063444  +0.00634  +0.0063444 

-0.0071898  -0.0071952  -0.00719  -0.0071953 

-0.23469  -0.2342416  -0.2342424 


NOTES. 

In  the  "New  Theory,"  Page  511,  HILL  points  out  that  the  convergence  of  the 
expansion  of  the  perturbing  function  is  slow  in  this  case,  the  terms  of  the  fifth  order 
in  the  inclinations  and  eccentricities  amounting  to  one  per  cent,  of  those  of  the  first 
order.  He  stated  that  a  computation  by  GAUSS'S  method  would  be  very  desirable 
and  consequently  this  was  effected  by  DR.  ASAPH  HALL,  JR.,  in  July,  1891  (A.  J. 
No.  244),  and  by  INNES  in  November,  1891  (M.  N.,  Vol.  LII,  Nos.  2  and  7).  HALL'S 
computation  is  the  first  application  of  GAUSS'S  method  made  after  the  publication 
of  HILL'S  memoir. 

Both  HALL  and  INNES  employed  the  values  of  the  elements  stated  by  LEVERRIER; 
Hall  divided  the  orbit  of  the  Earth  into  twelve  parts  and  INNES  into  sixteen.  The 
values  of  [dpldt]00  and  [dq/dt]0Q  given  by  the  latter  were  however  in  error  owing  to  a 
misprint  that  occurred  in  HILL'S  original  paper  in  the  value  of  J3]  in  M.  N.,  Vol.  LII, 
No.  7,  INNES  pointed  out  this  error  but  did  not  re-compute  the  variations. 

The  final  results  of  the  present  paper  were  printed  in  A.  J.,  No.  518,  but  the 
values  there  given  are  all  slightly  incorrect  owing  to  errors  in  some  of  the  preliminary 
constants,  which  remained  undetected  even  in  the  duplication.  Upon  devising  new 
test  equations  these  were  always  applied  to  all  computations  previously  made  and  in 
this  way  the  errors  affecting  practically  every  figure  of  the  present  computation  were 
discovered.  The  work  was  then  both  repeated  and  duplicated  so  that  it  is  hardly 
possible  that  any  errors  can  yet  remain  in  it. 

The  latter  part  of  INNES'  computation  was  also  duplicated,  the  values  of  J3,  WQ, 
[dp/dt]0  and  [dq/dt]0  being  freed  from  the  errors  referred  to  by  him.  It  is  these  corrected 
values  which  are  given  above. 

It  will  be  noticed  that  the  agreement  of  the  results  here  given  with  those  of 
HALL  is  very  exact  notwithstanding  the  difference  of  the  original  elements  used  in 
the  computation.  The  divergences  from  those  of  INNES  are  more  considerable, 
probably  because  the  latter  computer  did  not  employ  the  accurate  tables  of  HILL. 


138 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


The  values  obtained  in  the  "New  Theory  "  for  the  motion  of  the  plane  of  the 
ecliptic  are, 

[dp/dt]m  =  +0.0063362 

[dq/dt]^  =  -0.0072112 


ACTION  OF  JUPITER  ON  THE  EARTH. 

E 

A 

B  cos  t 

B  sin  t 

g 

0° 

28.04923872 

+  1.4444306 

+5.1077212 

1.6444314 

30 

27.80097551 

-1.1738070 

+4.4818820 

1.2661417 

60 

27.62449781 

-3.1281533 

+2.6317812 

0.4365772 

90 

27.56719541 

-3.8949449 

+0.0531514 

0.0001781 

120 

27.64447424 

-3.2687173 

-2.5630644 

0.4140764 

150 

27.83557589 

-1.4172691 

-4.5158555 

1.2854086 

180 

28.08919180 

+  1.1633052 

-5.2819683 

1.7585429 

210 

28.33731437 

+3.7815432 

-4.6561285 

1.3665053 

240 

28.51351085 

+5.7358901 

-2.8060283 

0.4963014 

270 

28.57067253 

+6.5026807 

-0.2273984 

0.0032594 

300 

28.49353432 

+5.8764527 

+2.3888176 

0.3596892 

330 

28.30271399 

+4.0250047 

+4.3416085 

1.1881266 

2: 

168.41444773* 

+7.82320801 

-0.52274091 

5.1096185 

Si 

168.41444770 

+7.8232076 

-0.5227406 

5.1096197 

E 

I 

G 

G' 

G" 

O 

0 

0.977739 

27.0061276 

1.0387013 

0.0586223 

30 

0.729882 

27.0062768 

0.7909419 

0.0592752 

60 

0.554312 

27.0065429 

0.5826671 

0.0277442 

90 

0.497498 

27.0066658 

0.4975109 

0.0000133 

120 

0.574342 

27.0065199 

0.6004570 

0.0255347 

150 

0.764478 

27.0062522 

0.8240511 

0.0577594 

180 

1.017523 

27.0061314 

1.0803044 

0.0602760 

210 

1.266039 

27.0062772 

1.3067275 

0.0387224 

240 

1.443204 

27.0065561 

1.4565395 

0.0126169 

270 

1.500952 

27.0066843 

1.5010367 

0.0000804 

300 

1.423446 

27.0065354 

1.4332595 

0.0092925 

330 

1.231720 

27.0062551 

1.2681196 

0.0346927 

Si 

5.990566 

162.0384133 

6.1919288 

0.1940866 

22 

5.990569 

162.0384114 

6.1883877 

0.1905434 

*  6o2  +  3aV  +  6[a'2  - 

Zkaa'ee'  cos  K]  =  168.41444773, 

f  6[a'  e'  —  kaa'e  r.os  K] 

=  +  7.8232074. 

- 

|  —  Gk'aa'  cos  <f>'  •  e  sin 

K'  -  -  0.5227409. 

h 

27.008467 

27.008061 

27.007154 

27.006666 

27.007100 

27.008066 

27.008637 

27.008243 

27.007275 

27.006689 

27.007056 

27.007962 

162.045689 

162.045687 

O 

e 

/      // 

11 

36  58.64 

10 

12  32.08 

8 

38  31.93 

7 

48   2.72 

8 

45  10.87 

10 

23  57.24 

11 

50  45.30 

12 

53  16.35 

13 

29   4.47 

13 

38  11.24 

13 

21  37.74 

12 

40  46.32 

67 

42   8.95 

67 

36  45.94 

OF   THE    ORBITS   OF   THE    FOUR   INNER    PLANETS. 


139 


ACTION  OF  JUPITER  ON  THE  EARTH. 

E 

logtfo 

log  Lo'         log  N,         log  N         log  P 

logQ 

0° 

0.01351621 

0.29098220    0.19630730    7.8502195    5.2763937 

6.6141228 

30 

0.01041684 

0.28686620    0.19168189    7.8490834    5.2711158 

6.6083484 

60 

0.00744942 

0.28292144    0.18724767    7.8522627    5.2713543 

6.6075955 

90 

0.00606351 

0.28107775    0.18517479    7.8588567    5.276992T 

6.6125603 

120 

0.00764272 

0.28317853    0.18753669    7.8670773    5.2864976 

6.6227349 

150 

0.01081185 

0.28739101     0.19227173    7.8747484    5.2973550 

6.6346279 

180 

0.01406123 

0.29170555    0.19712004    7.8798617    5.3067059 

6.6445510 

210 

0.01667410 

0.29517157    0.20101379    7.8810685    5.3120661 

6.6499953 

240 

0.01827428 

0.29729273    0.20339625    7.8780194    5.3119679 

6.6497435 

270 

0.01869373 

0.29784858    0.20402051    7.8714850    5.3063886 

6.6440330 

300 

0.01793523 

0.29684340    0.20289160    7.8631935    5.2968004 

6.6344668 

330 

0.01613325 

0.29445437    0.20020816    7.8553923    5.2858029 

6.6235786 

Si 

0.07887909 

1.74292385     1.17449955    7.1906339    1.7497196 

9.7732144 

22 

0.07879328 

1.74280948     1.17437087    7.1906342    1.7497203 

9.7731434 

E 

logF 

Ji'            Ji            J, 

F2 

0° 

6.6129514 

27.064741582     +0.24634548     +0.02063093 

-6.6623169 

30 

6.6071626 

27.061450021     +0.21012324     +0.32748544 

-5.8459973 

60 

6.6070396 

27.022786983     +0.12128740     +0.54805936 

-3.4327952 

90 

6.6125600 

26.992014082     +0.00316191     +0.62325071 

-0.0693287 

120 

6.6222233 

27.021540830     -0.11740027     +0.53291210 

+3.3431631 

150 

6.6334726 

27.060747670     -0.21222668     +0.30124976 

+  5.8903093 

180 

6.6433468 

27.066395282     -0.25546365     -0.00966336 

+6.8895984 

210 

6.6492222 

27.041108427     -0.23112984     -0.31651859 

+6.0732768 

240 

6.6494917 

27.008005002     -0.14158677     -0.53709407 

+3.6600761 

270 

6.6440314 

26.992057684     -0.01087596     -0.61228634 

+0.2966099 

300 

6.6342813 

27.004932957     +0.12157000     -0.52194700 

-3.1158827 

330 

6.6228858 

27.037481542     +0.21569955     -0.29028297 

-5.6630276 

Zi 

9.7693340 

162.188402634*    -0.02524781     +0.03289796 

+0.6818428 

S2 

9.7693345 

162.184859426     -0.02524778     +0.03289801 

+0.6818424 

*  s,(J,'  -  G")  =  161 

.994316084. 

z»l/i'  -  G")  =  161 

,994316076. 

140 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  JUPITKK  ON  THE  EARTH. 

E 

F, 

1000  X  fio     1000  X  So       1000  X  W0     1000  X  B(n) 

1000  X  S(n) 

0° 

-0.14994500 

3.5906058   -0.02485742   +0.00562843     0.0000000 

-0.02528143 

30 

-0.11198359 

3.5669877   -0.02409495   +0.13045170   +1.8097792 

-0.02445007 

60 

-0.03628181 

3.5812719   -0.01504685   +0.22107406   +3.1277000 

-0.01517409 

90 

+  0.00006601 

3.6318358   -0.00001620   +0.25540271   +3.6318358 

-0.00001620 

120 

-0.04119166 

3.7069905   +0.01547114   +0.22249821   +3.1836522 

+0.01534248 

150 

-0.12070192 

3.7863426   +0.02555514   +0.12714491   +1.8660682 

+0.02518930 

180 

-0.16035007 

3.8464128   +0.02722801   -0.00750003    0.0000000 

+0.02677889 

210 

-0.12100025 

3.8690539   +0.02153453   -0.14361296   -1.9068325 

+0.02122624 

240 

-0.04149417 

3.8484336   +0.01189802   -0.24048156   -3.3051260 

+0.01179908 

270 

+0.00005467 

3.7925083   +0.00121404   -0.26976404   -3.7925083 

+0.00121404 

300 

-0.03599894 

3.7183323   -0.00934064   -0.22557067   -3.2473996 

-0.00941964 

330 

-0.11169659 

3.6453213   -0.01884026   -0.12397381   -1.8495240 

-0.01911793 

s, 

-0.46526165 

22.2920469   +0.00535226   -0.02435156   -0.2411734 

+0.00404529 

S2 

-0.46526167 

22.2920496   +0.00535230   -0.02435149   -0.2411816 

+0.00404538 

E 

•SSSL. 

1000  X|-#i>  cos  v 
..  1000  XWa  cos  (»+»•)   1000XW0sin(y+ir) 

OJ     /  f            \  '    r»  1 

+  (  sec-  if  +  1  1  sin  r<So 
\o       /     J 

1000X-2-Ro 
a 

0° 

-0.0497148 

-3.5906058     -0.00101227     +0.00553665 

-  7.0607738 

30 

+  1.7678935 

-3.0981989     -0.08531744     +0.09868428 

-  7.0303590 

60 

+3.1124039 

-1.7713812     -0.20927556     +0.07125641 

-  7.1024820 

90 

+3.6313250 

+0.0608776     -0.25043240     -0.05014132 

-  7.2636717 

120 

+3.1675401 

+  1.9264211     -0.16743708     -0.14652740 

-  7.4761534 

150 

+  1.8214372 

+3.3200889     -0.04173395     -0.12010043 

-  7.6826719 

180 

-0.0544560 

+3.8464128     -0.00134888     +0.00737774 

-  7.8218420 

210 

-1.9439518 

+3.3453092     -0.09208972     +0.11020057 

-  7.8505000 

240 

-3.3167069 

+  1.9516989     -0.22529979     +0.08409148 

-  7.7614089 

270 

-3.7919952 

+0.0611766     -0.26614166     -0.04405982 

-  7.5850167 

300 

-3.2561646 

-1.7957525     -0.17399247     -0.14355750 

-  7.3742992 

330 

-1.8818157 

-3.1224540     -0.04265147     -0.11640599 

-  7.1847533 

Zi 

-0.3970983 

+0.5667933     -0.77836605     -0.12182259 

-44.5969593 

22 

-0.3971070 

+0.5667994     -0.77836664     -0.12182271 

-44.5969726 

sin  <f  •  %A  i(*>  +  cos  if 

•  Bow  =  -  0.0000000000093. 

OF   THE    ORBITS   OF   THE   FOUR   INNER   PLANETS.  141 

DIFFERENTIAL  COEFFICIENTS. 

n  log  coeff. 

[de/dilw    =  -     85.760340  m'  n  1.9332865 

[dx/dt]ao  =  [dv/dilw    =  +7298.7450     TO'  p  3.8632482 

[dp/dt]m    =  -       26.316855  TO'  n  1.4202340 

[dq/dt]m    =  -     168.14734    TO'  n  2.2256900 

[dL/dt]w   =  -9631.7202      TO'  n  3.9837038 

FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 

[de/dt]m    =  -0^081841849 
[dx/dt]w  =  [dTr!dt]w    =  +6.9652565 

[dp/dt]w    =  -0.025114405 

[dq/dt]m    =  -0.16046446 

[dLfdt]w  =  -9.1916336 
COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcoml).  Method  of  Gauss. 


[de/dt]oo 

-0.08182 

-0.08182 

-0.0818418 

e[dir/dl]00 

+0.11679 

+0.11677 

+0.1168153 

[dp/dtlw 

-0.02501 

-0.02511 

-0.0251144 

[dq/dt]m 

-0.16041 

-0.16047 

-0.1604644 

[dL/dt]m 

-9.1916 

-9.1916336 

NOTES. 

The  very  close  agreement  of  the  sums  toward  the  end  of  this  computation  is 
owing  to  the  circularity  of  the  two  orbits  and  to  their  small  mutual  inclination.  It  is 
evident  that  a  division  into  eight  parts  would  have  been  sufficient,  while  the  errors 
arising  from  a  division  into  only  six  parts  would  have  been  almost  inappreciable. 

In  this,  as  in  several  other  cases,  the  divergence  from  the  last  figure  of  NEW- 
COMB'S  results  is  rather  larger  than  was  to  have  been  expected.  The  values  stated 
by  NEWCOMB  were  computed  to  one  more  significant  figure  than  was  published  to 
insure  the  accuracy  of  the  final  figure  given.  The  uncertainty  of  this  figure  is  evi- 
dently due  to  neglected  terms  in  the  series  employed  by  LEVERRIER  and  NEWCOMB. 
In  the  present  case  we  obtain  for  [deldt]00, 

Computed  from  the  six  even  points  of  division —  0".0818428 

Computed  from  the  six  odd  points  of  division   —0  .0818409, 


142 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


and  the  difference  between  any  two  corresponding  values  for  any  other  coefficient  is 
even  less  than  this. 

The  values  of  the  coefficients  which  define  the  motion  of  the  plane  of  the  ecliptic 
are  stated  by  HILL  as  follows:  „ 

[dp/dt]^    =  -0.0251149 

[dq/dt]w    =  -0.1604628 


E          A 

0°  92.9909218 

30  92.7594069 

60       92.3168471 

90  91.7819295 

120      91.2980364 

150  90.9947748 

180  90.9533006 

210  91.1846750 

240  91.6269534 

270  92.1617302 

300  92.6457640 

330  92.9493070 

Zj  551.8318232* 

22  551.8318234 

E          I 

0  +2.001263 

30  +1.768419 

60  +1.323475 

90  +0.787549 

120  +0.304998 

150  +0.004086 

180  -0.036365 

210  +0.193697 

240  +0.633602 

270  +1.167320 

300  +1.652663 

330  +1.958598 

21  +5.879636 

22  +5.879669 


t  6[a'V  -  fcao'e  cos  K]  =  +  29.6608842. 
t  -  6k'aa'  cos  <p'  •  e  sin  K'  =  -  0.1707000. 


ACTION  OF  SATURN  ON  THE  EARTH. 

B  cos  . 

B  sin  t 

d 

+14.3293908 

+  1.6679163 

0.795517 

+  12.2250618 

+6.1218324 

10.716773 

+  8.1696352 

+8.9277816 

22.792314 

+  3.2497638 

+9.3339213 

24.913194 

•  1.2162762 

+7.2314183 

14.953684 

-  4.0318145 

+3.1836393 

2.898337 

-  4.4424293 

-1.7248166 

0.850721 

-  2.3380978 

-6.1787321 

10.916915 

+  1.7173265 

-8.9846857 

23.083782 

+  6.6371970 

-9.3908217 

25.217872 

+  11.1032373 

-7.2883175 

15.189927 

+  13.9187769 

-3.2405388 

3.002863 

+29.6608843f 

-0.1707036t 

77.665945 

+29.6608873 

-0.1706996 

77.665954 

G 

G' 

G" 

90.703603 

2.0057342 

0.0043727 

90.703703 

1.8341642 

0.0644170 

90.704603 

1.4944314 

0.1681447 

90.705370 

1.0517493 

0.2611465 

90.705257 

0.5874560 

0.2806341 

90.704380 

0.1809884 

0.1765509 

90.703605 

0.0803976 

0.1166593 

90.703690 

0.4578844 

0.2628566 

90.704569 

0.9146652 

0.2782376 

90.705348 

1.3729264 

0.2025015 

90.705263 

1.7502260 

0.0956818 

90.704378 

1.9757281 

0.0167564 

544.226899 

6.8329103 

0.9437302 

544.226868 

6.8734407 

0.9842289 

'ee'  cos  K]  =  551.8318229. 

h 

90.703702 

90.705031 

90.707414 

90.708424 

90.707081 

90.704732 

90.703708 

90.705020 

90.707394 

90.708453 

90.707143 

90.704751 

544.236442 

544.236411 

e 

O 

i   n 

8 

33  39.757 

8 

18  56.347 

7 

46  25.450 

6 

53  59.891 

5 

36  19.643 

3 

35  46.006 

2 

40  11.401 

5 

6  24.385 

6 

34  30.181 

7 

33  52.413 

8 

11  49.887 

8 

31  21.103 

39 

22  56.319 

40 

0  20.145 

OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


143 


ACTION  OF 

SATUKN  ON 

THE 

EARTH. 

E 

logKo 

log  La 

logA^o 

log  AT 

logP 

logQ 

0° 

0.00730944 

0 

.28273527 

0 

.18703836 

7. 

0561506 

3.4235951 

5.2855436 

30 

0.00689441 

0 

.28218320 

0 

.18641769 

7.0572862 

3.4236026 

5.2857705 

60 

0.00602140 

0 

.28102171 

0 

.18511178 

7.0610565 

3.4252108 

5.2877346 

90 

0.00473960 

0 

.27931572 

0 

.18319348 

7.0664172 

3.4279697 

5.2907291 

120 

0.00312447 

0 

.27716505 

0 

.18077485 

7.0719166 

3.4311334 

5.2937173 

150 

0.00128428 

0 

.27471328 

0 

.17801715 

7. 

0761000 

3.4338677 

5.2956443 

180 

0.00070759 

0 

.27394462 

0 

.17715249 

7.0778799 

3.4354590 

5.2968496 

210 

0.00259230 

0.27645617 

0 

.17997757 

7. 

0767946 

3.4354874 

5.2978905 

240 

0.00430241 

0 

.27873368 

0.18253896 

7.0731165 

3.4339314 

5.2966961 

270 

0.00570043 

0 

.28059458 

0 

.18463152 

7. 

0677983 

3.4311901 

5.2938284 

300 

0. 

00669846 

0.28192252 

0.18612461 

7.0622487 

3.4279906 

5.2902830 

330 

0.00724349 

0.28264754 

0 

.18693973 

7.0579727 

3.4252033 

5.2872040 

Zi 

0.02816377 

1 

.67552285 

1.09874105 

2.4023686 

6.1773201 

1.7508241 

2, 

0.02845451 

1 

.67591049 

1 

.09917714 

2.4023690 

6.1773208 

1.7510668 

E 

log  V 

J,' 

/, 

/i 

Ft 

0° 

5.2855175 

90.7004946 

+0.12827478 

-0.7968027 

-  8.486434 

30 

5.2853861 

90.7506534 

+0.29200243 

+  1.2416991 

-31.148158 

60 

5.2867312 

90.7742572 

+0.41637229 

+2.9535293 

-45.424968 

90 

5.2891707 

90.7991434 

+0.49225830 

+3 

.8800058 

-47.491413 

120 

5.2920417 

90.8302700 

+0.45906255 

+3.7728791 

-36.793788 

150 

5.2945889 

90.8046167 

+0.26449000 

+2 

.6608537 

-16.198502 

180 

5.2961519 

90.8127812 

-0.06165231 

+0.8418949 

+  8.775944 

210 

5.2963206 

90.9507883 

-0.39625124 

-1 

.1966086 

+31.437674 

240 

5.2950355 

90.8893132 

-0.59035199 

-2 

.9084413 

+45.714495 

270 

5.2926201 

90.7428396 

-0.56521208 

-3 

.8349184 

+47.780934 

300 

5.2897120 

90.6423828 

-0.36124653 

-3 

.7277897 

+37.083286 

330 

5.2871040 

90.6419564 

-0.09682848 

-2 

.6157627 

+  16.488008 

Si 

1.7451898 

544.6494990* 

-0.00954121 

+0.1352696 

+  0.868535 

22 

1.7451904 

544.6899978 

-0.00954107 

+0.1352689 

+  0.868543 

*2,(J 

i'  -  G")  = 

543.7057688. 

zs(J 

i'  -  G")  = 

543.7057689. 

144 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


E 

0° 
30 
60 
90 
120 
150 
180 
210 
240 
270 
300 
330 

Si 
E 

ACTION  OF  SATURN  ON  THE  EARTH. 
F,        1000  X  Ro      100000  X  So     100000  X  Wo     10000  X  RM     100000  X  S<»> 

-0.3555837   0.57362783    +0.02247623    -1.5471185     0.0000000    +0.0228596 
-1.2681621   0.57437868    -0.26276188    +2.3619011    +2.9142195    -0.2666346 
-1.2912829   0.57674453    -0.40344654    +5.6791610    +5.0370180    -0.4068583 
-0.3956591   0.58127800    -0.31429014    +7.5403759    +5.8127800    -0.3142901 
+0.5220384   0.58774791    -0.09358683    +7.4052797    +5.0477180    -0.0928086 
+0.5361282   0.59426527    +0.08130415    +5.2579217    +2.9287882    +0.0801402 
-0.3802586   0.59796705    +0.11726800    +1.6546186     0.0000000    +0.1153337 
-1.3248861   0.59688219    +0.07295990    -2.4035193    -2.9416851    +0.0719154 
-1.3648579   0.59139747    +0.07709136    -5.7742147    -5.0790610    +0.0764503 
-0.4663692   0.58408733    +0.18082475    -7.5353293    -5.8408733    +0.1808248 
+0.4731390   0.57795107    +0.28959440    -7.2511167    -5.0475302    +0.2920434 
+0.5221422   0.57454813    +0.25136029    -5.0525337    -2.9150792    +0.2550648 

-2.3968057   3.50543586    +0.00939762    +0.1666094    -0.0418552    +0.0070201 
-2.3968061   3.50543960    +0.00939707    +0.1688164    -0.0418499    +0.0070205 

• 

lOOOXtft  sinv    1000  xT-flo  cos  w 

+  (coav+cosE)X0}  _ 

if       \ 

h  (  ~  sec2  ^+  1  I  sin  t'o'c 

J 

a 

0° 

+0.00044952 

-0.57362783 

+0.002782496 

-0.015218912 

-0.0011280149 

30 

+0.28684099 

-0.49762977 

-0.015447198 

+0.017867337 

-0.0011320724 

60 

+0.49964759 

-0.28807540 

-0.053760700 

+0.018305034 

-0.0011438222 

90 

+0.58124893 

+0.00316289 

-0.073936353 

-0.014803463 

-0.0011625560 

120 

+0.50564826 

+0.29959114 

-0.055727122 

-0.048767876 

-0.0011853529 

150 

+0.29142597 

+0.51791202 

-0.017258560 

-0.049666046 

-0.0012057929 

180 

-0.00234536 

+0.59796705 

+0.002975836 

-0.016276382 

-0.0012159913 

210 

-0.29539381 

+0.51865750 

-0.015412218 

+0.018443264 

-0.0012111028 

240 

-0.50861524 

+0.30174597 

-0.054096837 

+  0.020191247 

-0.0011927133 

270 

-0.58403554 

+0.00617932 

-0.074341448 

-0.012307244 

-0.0011681747 

300 

-0.50182267 

-0.28668159 

-0.055931013 

-0.046147489 

-0.0011462092 

330 

-0.28712395 

-0.49766091 

-0.017382540 

-0.047441087 

-0.0011324063 

S, 

-0.00703790 

+0.05091934 

-0.213757340 

-0.087914378 

-0.0070121038 

S, 

-0.00703741 

+0.05092105 

-0.213778317 

-0.087907239 

-0.0070121051 

sin  <p  •  yAiM  +  cos  <f  • 

Bo(e>  =  +  0.00000000000028. 

OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  145 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[de/dtlw    =  -       1.5163927  TO'  n  0.1808117 

[dx/dt]m  =  [dT/dfloo    =  +  655.70924      TO'  p  2.8167113 

[dp/dt]m    =  -       18.991017    TO'  n  1.2785482 

[dq/dt]m    =  -       46.179399    TO'  n  1.6644483 

=  -1514.4911        TO'  n  3.1802667 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF 

[de/dt]M    =  -0.00043305713 
[dx/df]«,  =  [dir/d4»    =  +0.18725991 

[dp/dfloo    =  -0.0054235259 
[dq/dt]M    =  -0.013188086 
=  -0.43251400 


m 


COMPARISON  WITH  OTHER  RESULTS. 

Lcverrier.  Newcomb.  Method  of  Gauss. 

[de/dt]w  -0.00044  -0.00043  -0.00043306 

e[dirldt}w  +0.00315  +0.00314  +0.00314056 

[dp/dt]m  -0.00542  -0.00542  -0.00542353 

[dq/dt]M  -0.01317  -0.01318  -0.01318809 

[dL/dt]M  -0.4325  -0.43251400 

NOTES. 

Here,  as  in  the  previous  case,  the  approximate  tests  completely  fail  with  the 
angle  e,  the  roots  G,  G',  G"  ,  and  with  the  functions  which  immediately  depend  upon 
these  quantities.  The  close  agreement  of  the  final  sums  shows,  however,  that  the 
expansion  of  the  perturbing  function  is  quite  rapidly  convergent  for  this  case. 

The  values  obtained  by  HILL  in  the  "New  Theory  "  are: 


[dp/dt]w  =  -oo054237         [dq/dt]^  =  -0.0131883 
The  agreement  of  the  final  results  here  obtained  with  all  other  values  is  satisfactory. 


146 


THE    SECULAR   VARIATIONS    OF   THE   ELEMENTS 


ACTION  OF  URANUS  ON  THE  EARTH. 


E 

A 

B  cos  t            B  sin  e             g 

h 

0° 

369.9391833 

+24.383407      -17,162615      247.29194 

367.49698 

45 

370.9299673 

+34.837638       -  7.159449      41.56718 

367.49556 

90 

370.9628887 

+34.937206      +  7.512108      45.76306 

367.49557 

135 

370.0188613 

+24.623780      +17.957661      261.51163 

367.49708 

180 

368.6506847 

+  9.938828      +18.058344      264.45220 

367.49706 

225 

367.6596194 

-  0.515405      +  7.755178      48.77249 

367.49561 

270 

367.6264169 

-  0.614971       -  6.916382      38.79263 

367.49553 

315 

368.5707253 

+  9.698453      -17.361932      244.44864 

367.49688 

Si 

1477.1791736* 

+68.644470f     +  1.191455t     596.29983 

1469.98514 

Z2 

1477.1791733 

+68.644466      +  1.191458      596.29994 

1469.98513 

E 

/ 

G            G'           G" 

e 

O 

O 

i   a 

0 

+  1.63126 

367.495141      1.9739873      0.3408899       4 

33   0.174 

45 

+2.62346 

367.495250      2.6661957      0.0424235       4 

55  28.953 

90 

+2.65637 

367.495229      2.7027882      0.0460735       4 

57  40.407 

135 

+  1.71085 

367.495130      2.0584810      0.3456945       4 

38  13.695 

180 

+0.34269 

367.495095      1.0379451      0.6933002       3 

55  55.015 

225 

-0.64694 

367.495249      0.163776T      0.8103506       2 

56  52.565 

270 

-0.68006 

367.495243      0.1303078      0.8100787       2 

53  46.998 

315 

+0.26290 

367.495069      0.9586081      0.6938970       3 

50  28.866 

S, 

+3.95026 

1469.980708      5.8450284      1.8903423      16 

20  22.594 

22 

+3.95027 

1469.980698      5.8470609      1.8923656      16 

21   4.079 

ACTION  OF  URANUS  ON  THE  EARTH. 

E 

log  A'0 

log  LO'        log  No        log  N       log  P 

logQ 

0° 

0.00205713 

0.27574316    0.17917560    6.1388849     1.2833195 

3.7524062 

45 

0.00241015 

0.27621390    0.17970508    6.1440956     1.2897058 

3.7584989 

90 

0.00244638 

0.27626178    0.17975893    6.1514873     1.3001369 

3.7689401 

135 

0.00213675 

0.27584925    0.17929492    6.1638868     1.3084161 

3.7775217 

180 

0.00153562 

0.27504825    0.17839395    6.1668770     1.3097851 

3.7792009 

225 

0.00086277 

0.27415148    0.17738519    6.1617903     1.3035252 

3.7729672 

270 

0.00083285 

0.27411159    0.17734031     6.1515209     1.2932165 

3.7626532 

315 

0.00146560 

0.27195493    0.17828898    6.1419973     1.2848106 

3.7542155 

S, 

0.00687198 

1.10116478    0.71466879    4.611770(1    5.1864579 

3.0632003 

22 

0.00687557 

1.10116956    0.71467417    4.6117700    5.1864577 

5.0632033 

*  4a2  +  2aV  +  4[o'2  — 

2kaa'ee'  cos  A:]  =  1477.1791732. 

t  4[a'V  -  kaa'e  cos  K]  =  +  68.644468. 
t  +  4fc'aa'  cos  v'  •  e  sin  K'  =  +  1.191454. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


147 


ACTION  OF  URANUS  ON  THE  EARTH 

. 

E 

logF 

Ji'                             J* 

Ji 

Fl 

0° 

3.7519032 

367.8221780             -0.84734235 

+2.2088876 

+301.43340 

45 

3.7584363 

367.4766488             -0.35515557 

+4.6641774 

+  123.58384 

90 

3.7688721 

367.4892672             +0.38057614 

+4.3579828 

-129.67133 

135 

3.7770116 

367.8350259             +0.86234922 

+  1.4696662 

-309.97871 

180 

3.7781780 

368.1745883              +0.82112849 

-2.3088335 

-311.71664 

225 

3.7717715 

368.2455011              +0.34414218 

-4.7641198 

-133.86711 

270 

3.7614579 

368.2514548             -0.29842856 

-4.4579225 

+  119.38809 

315 

3.7531918 

368.1823028             -0.79535630 

-1.5696087 

+  299.69549 

Si 

5.0604111 

1471.7374883*           +0.05593372 

-0.1998856 

-  20.56648 

22 

5.0604112 

1471.7394786             +0.05597953 

-0.1998849 

-  20.56649 

E 

r, 

1000000  X  Ra        1000000  X  So          1000000  X  W, 

1000000  X  -R<"> 

1000000  X  S<"> 

0° 

+3.5560766 

68.949325        +0.10018820        +1.2544323 

0.000000 

+0.10189713 

45 

+0.4865778 

69.825413        +0.03716997        +2.6752556 

+49.966575 

+0.03761606 

90 

+0.8230258 

71.523975        -0.03529126        +2.5611571 

+71.523975 

-0.03529126 

135 

+4.0393764 

73.045932        -0.11453773        +0.8877061 

+51.045918 

-0.11319535 

180 

+3.8028425 

73.443068        -0.14342145        -1.3776247 

0.000000 

-0.14105578 

225 

+0.5809413 

72.487200        -0.06580265        -2.8156195 

-50.655465 

-0.06503154 

270 

+0.709709S 

70.794418        +0.06221357        -2.5724979 

-70.794418 

+0.06221357 

315 

+3.7847601 

69.352325        +0.12685775        -0.8818747 

-49.628040 

+0.12838021 

Si 

+8.8916547 

284.710786        -0.01631094        -0.1345332 

+  0.729557 

-0.01223634 

2* 

+8.8916556 

284.710870        -0.01631266        -0.1345325 

+  0.728988 

-0.01223053 

1  0ftOOOn  V  1  7?n  rct<*  u 

E 

1  000000  X[R0  sin  v 

XL                          1000000                          1000000 

1000x-2r-K0 

+  (cosv+cosE)Sl 

l]     +^Bec'«>+l)<rfa»S.]    X^°cos  ("+"•) 

KWosm(v-\-ir) 

a 

0° 

+  0.200376 

-68.949325              -0.2256099 

+  1.2339773 

-0.13558593 

45 

+50.011805 

-48.728590              -2.2190515 

+  1.4942570 

-0.13799470 

90 

+71.514500 

+  1.128956              -2.5113156 

-0.5028130 

-0.14304794 

135 

+  51.201671 

+52.095595              -0.4959295 

-0.7362581 

-0.14782435 

180 

+  0.286843 

+73.443068              -0.2477661 

+  1.3551611 

-0.14934959 

225 

-50.554733 

+51.949429              -2.2975222 

+  1.6276075 

-0.14669365 

270 

-70.785500 

+  1.062876              -2.5379547 

-0.4201590 

-0.14158884 

315 

-49.442739 

-48.631449              -0.5098809 

-0.7195307 

-0.13705976 

ft 

+  1.216219 

+  6.685575              -5.5226463 

+  1.6661664 

-0.56957230 

22 

+  1.216004 

+  6.684985             -5.5223841 

+  1.6660757 

-0.56957246 

sin  <p  •  \Ai(*>  +  cos  . 

,p  .  J30<c>  =  -  0.00000000000025. 

*  2,(J,'  -  G")  =  1469.8471460. 

2«(Ji'  -  G")  =  1469.8471130. 

148  THE  SECULAR  VARIATIONS  OF  THE  ELEMENTS 

DIFFERENTIAL  COEFFICIENTS. 

n  log  coeff. 

[<fe/«tt]oo    =  +     0.39395664  TO'  p  9.5954484 

[dxfdtlw,  =  [drldt]M    =  +129.13143       TO'  p  2.1110320 

[dp/dt]w    =  +     0.53988815  TO'  p  9.7323038 

[dq/dt]w    =  -       1.7895101    TO'  n  0.2527342 

[dL/(ft]oo   =-184.51950       TO'  n  2.2660422 

FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF 


TO 


=  +0.000017278801 
[dx/dt]M  =  [Ar/dfloo    =  +0.0056636605 
=  +0.000023679306 
=  -0.000078487295 
=  -0.0080929604 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

[(fe/tft]oo    +0^00002  +0^00002  +o!oOOO  172788 

e[d7r/d<]oo   +0.00009  +0.00010  +0.0000949860 

[dpldt]m    +0.00002  +0.00002  +0.0000236793 

[dq/dt]m    -0.00008  -0.00008  -0.0000784873 

-0.0081  -0.0080929604 


NOTES. 

It  will  be  noticed  that,  owing  to  the  very  small  mutual  inclination,  the  approxi- 
mate tests  are  here  more  exactly  satisfied  than  even  in  the  case  of  Saturn,  where 
twelve  points  of  division  were  employed.  It  is  therefore  evident  that  eight  points 
are  fully  sufficient  and  that  the  greatest  error  arising  from  a  division  into  only  four 
points  (which  occurs  with  the  coefficient  [dir/dt]00),  could  not  amount  to  more  than 
1  /20,000th  of  the  whole. 

The  results  obtained  by  HILL  are  : 

[dpldt]M  =  +0.0000237        [dqldt]m  =  -0.0000785 
exactly  agreeing  with  those  here  given. 


OP   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


149 


ACTION  OF  NEPTUNE  ON  THE  EARTH. 


E 

A 

0° 

905.47785591 

45 

905.10315254 

90 

904.80486595 

135 

904.75792710 

180 

904.98963355 

225 

905.36405558 

270 

905.66206098 

315 

905.70928102 

Zi 

3620.93441639* 

22 

3620.93441624 

E 

I 

0 

0 

1.23952 

45 

0.86483 

90 

0.56594 

135 

0.51893 

180 

0.75119 

225 

1.12575 

270 

1.42319 

315 

1.47027 

2, 

3.97984 

22 

3.97977 

E 

log  A'0 

0° 

0.00047205    0 

45 

0.00035874    0 

90 

0.00022606    0 

135 

0.00019146    0 

180 

0.00031307    0 

225 

0.00044509    0 

270 

0.00052334    0 

315 

0.00053136    0 

2i 

0.00153452    1 

y 

0.00152666     1 

B  cos  e           B  sin  «             g 

h 

+23.748411      +24.815277      40.194861 

904.17306 

+  1.127297     +28.979673      54.817494 

904.17306 

-17.814345     +15.920217      16.543607 

904.17365 

-21.980762      -  6.713038       2.941510 

904.17372 

-  8.931313      -25.661837      42.984089 

904.17317 

+  13.689802      -29.826236      58.066960 

904.17304 

+32.631450      -1.6.766778      18.349806 

904.17361 

+36.797860     +  5.866476       2.246398 

904.17374 

+29.634203f      -  1.693121f     118.072363 

3616.69349 

+29.634196      -  1.693125     118.072362 

3616.69355 

G            G'           G" 

e 

O      1      it 

904.17301       1.274452       0.034882 

2  10  50.926 

904.17299       0.930076       0.065186 

1  54   4.344 

904.17363       0.596627       0.030667 

1  30  33.476 

904.17372       0.525125       0.006195 

1  23  20.558 

904.17312       0.809936       0.058696 

1  46  33.999 

904.17297       1.180229       0.054414 

2   7   3.526 

904.17359       1.437325       0.014120 

2  17  46.322 

904.17374       1.471958       0.001688 

2  18  49.394 

3616.69335       4.118340       0.138365 

7  45  44.723 

3616.69342       4.107388       0.127483 

7  43  17.822 

ACTION  OF  NEPTUNE  ON  THE  EARTH. 

og/V         log  Wo         logtf         logP 

logQ 

'363061     0.17679925     5.5513789     9.9124729 

2.7719098 

347955     0.17662931     5.5555725     9.9164864 

2.7759190 

330267     0.17643033    5.5658262    9.9265957 

2.7859899 

'325655    0.17637844    5.5760491    9.9367960 

2.7961727 

'341867    0.17656083    5.5803398    9.9411989 

2.8006208 

'359467    0.17675882    5.5762686    9.9373080 

2.7967497 

'369899     0.17687617     5.5661355     9.9273173 

2.786753T 

370968    0.17688820    5.5557902    9.9169945 

2.7764257 

405094     0.70666658     2.2636803     9.7075847 

1.1452735 

404045     0.70665477     2.2636803     9.7075848 

1.145267T 

*  4a"  +  2aV  +  4[a'2  -  2kaa'ee'  cos  K\  =  3620.93441628. 
t4[a'V-fraa'ecos  A]  =  +29.634198. 
|  -  4k'aa'  cos  *.'.  e  sin  A"'  =  -  1.693118. 


150 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  NEPTUNE  ON  THE  EARTH. 

E 

logF 

Ji'                                          Ji                                             Ja 

F2 

0° 

2.7718889 

903.9916758             +0.58828709             -13.723633 

-190.54625 

45 

2.7758798 

904.1719863             +0.01502688             +  7.632175 

-222.52292 

90 

2.7859714 

903.5306465             -0.23467770             +24.659009 

-122.24477 

135 

2.7961690 

903.3686400              +0.17398234              +27.382788 

+  51.54664 

180 

2.8005856 

904.0154903             +0.15962945             +14.207956 

+  197.04666 

225 

2.7967170 

904.1717548             -0.46647484               -  7.147853 

+229.02334 

270 

2.7867446 

903.5394237             -0.52719220             -24.174694 

+  128.74518 

315 

2.7764247 

903.3535982             +0.26287497             -26.898466 

-  45.04625 

2! 

1.145190o 

3615.0772363*            -0.01395336             +  0.968638 

+  13.00082 

22 

1.1451905 

3615.0659793             -0.01459065             +  0.968644 

+  13.00081 

E 

F, 

100000  X  Ro        10000000  X  So           100000  X  We          100000  X  #<"> 

10000000  X  -S("> 

0° 

-  5.0649535 

1.7793453       +0.19215285       -0.08120439           0.0000000 

+  0.19543047 

45 

-  6.5852856 

1.7972262        -0.17462602       +0.04549983       +1.2860826 

-0.17672179 

90 

-  1.8342882 

1.8362237       -0.24659977       +0.15062685       +1.8362237 

-0.24659977 

135 

-  0.4453113 

1.8790643       +0.15337661        +0.17125249       +1.3131267 

+0.15157902 

180 

-  5.4164244 

1.9016607        +0.27295016       +0.08971979           0.0000000 

+0.26844799 

225 

-  6.9853355 

1.8852158       -0.09387083        -0.04482073        -1.3174254 

-0.09277065 

270 

-  2.0485736 

1.8383392        -0.21373143        -0.14796412        -1.8383392 

-0.21373143 

315 

-  0.3483062 

1.7940628       +0.11988971        -0.16075330        -1.2838186 

+0.12132856 

Si 

-  14.3642397 

7.3555689       +0.00477181       +0.01117813        -0.0021155 

+  0.00354726 

22 

-  14.3642386 

7.3555691        +0.00476947       +0.01117829        -0.0020347 

+  0.00341514 

E 

100000X[«osin,      I00000x[-Rocos,           imm                            l(mQQ 
+  (eosv+cosE)St,\     .  (r            ,  ,\  •      r,  1       XWoCos  (W+JT)              XlFosin  (W+JT) 

—I-  I    —   ^i<l<-  ,n  -I-  1     1  Q1T1   JlX.,    1 

100000x-2-flo 

a 

v»             /          .1 

0° 

+0.0038431 

-1.7793453            +0.01460463            -0.07988026 

-  3.4990076 

45 

+  1.2834469 

-1.2580636            -0.03774087            +0.02541381 

-  3.5518258 

90 

+  1.8360068 

+0.0258636            -0.14769558            -0.02957146 

-  3.6724475 

135 

+  1.3107603 

+  1.3464279            -0.09567261            -0.14203579 

-  3.8026901 

180 

-0.0054590 

+  1.9016607            +0.01613612            -0.08825681 

-  3.8671076 

225 

-1.3159048 

+  1.3499919            -0.03657334            +0.02590924 

-  3.8151452 

270 

-1.8380449 

+0.0351057            -0.14597725            -0.02416657 

-  3.6766784 

315 

-1.2819528 

-1.2550750            -0.09294409            -0.13116027 

-  3.5455738 

2i 

-0.0036540 

+0.1832847            -0.26293208            -0.22187510 

-14.7152411 

22 

-0.0036504 

+0.1832812             -0.26293091             -0.22187301 

-14.7152409 

sin  if  •  \A  i  (>>  +  cos  <f 

.  B0«  =  +  0.0000000000000014. 

*  2,(J,'  -  G")  =  3614.9388718. 

2,(J,'  -  G")  =  3614.9384968. 

OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  151 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[de/dt]m     =     -0.011831221m'  n  8.0730296 

[dx/dt]m  =  [dTr/dt]^    =  +35.402545        TO'  p  1.5490345 

[dp/e&]oo    =  --  0.71895833    TO'  n  9.8567037 

[dq/dt]m    =  --  0.85200049    TO'  n  9.9304399 

[dL/dt]w   =  -47.671428        TO'  n  1.6782582 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO' 

[de/dt]w  =  -0.00000060056972 

[dx!dt]w  =  [dw/dt]w  =  +0.0017970838 

[dp/dtlao  =  -0.000036495344 

[dg/dt]m  =  -0.000043248757 

[dL/dt]m  =  -0.0024198698 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.                    Method  of  Gauss. 

[de/dt}00        0.00000  0.00000             -0.00000060057 

e[drldt]oo    +0.00003  +0.00003             +0.00003013915 

[dpldt]m    -0.00004  -0.00004             -0.00003649534 

[dg/dt]m    -0.00004  -0.00004             -0.00004324876 


NOTES. 

The  mutual  inclination  is  here  nearly  twice  as  great  as  in  the  case  of  Uranus,  and 
yet  the  convergence  of  the  perturbing  function  is  more  rapid  because  the  eccentricity 
of  Neptune  is  so  much  smaller  than  that  of  Uranus.  Hence,  although  the  sums  of 
e,  G,  G',  G",  etc.,  are  in  great  disagreement,  the  final  sums  from  which  the  differential 
coefficients  are  obtained  are  almost  identical.  The  greatest  error  arising  from  a  divi- 
sion into  only  four  parts  occurs  with  the  coefficient  [dp/dt]00  and  amounts  to  but 
0".  000000000002 

The  results  of  HILL  are  : 


=  -00000366        [dqldt]n  =  -0.0000435 


152 


THE    SECULAR   VARIATIONS    OF   THE    ELEMENTS 


E 

0° 

30 

60 

90 

120 

150 

180 

210 

240 

270 

300 

330 


MARS. 

ACTION  OF  MERCURY  ON  MARS. 

A 

B  sin  ( 

B  COS  e 

1000  x  g 

2.01395126 

-0.51051059 

-0.07779H4 

1.6508943 

2.19162773 

-0.49427471 

+0.22547307 

1.5475600 

2.44453125 

-0.33152771 

+0.47957033 

0.6962245 

2.71228886 

+0.06587754 

+0.61641571 

0.0274906 

2.92685114 

+0.23149508 

+0.59934132 

0.3394636 

3.02703005 

+0.48090911 

+0.43292228 

1.4649936 

2.97859089 

+0.61553465 

+0.16175058 

2.4000193 

2.79081655 

+0.59929870 

-0.14151362 

2.2750785 

2.51771727 

+0.43655177 

-0.39561101 

1.2072061 

2.23986183 

+0.17090155 

-0.53245622 

0.1850128 

2.03539750 

-0.12647099 

-0.51538178 

0.1013191 

1.95541421 

-0.37588517 

-0.34896288 

0.8949940 

14.91703931* 

+0.31507221f 

+0.25187830J 

6.9351269 

14.91703923 

+0.31507194 

+0.25187834 

6.9351295 

1.87197670 
2.04795479 
2.29696077 
2.56269605 
2.77862654 
2.88207295 
2.83609064 
2.64795925 
2.37176681 
2.09074126 
1.88662892 
1.81063180 

14.04205038 
14.04205609 


G' 


G" 


0 

0.13564010 

1.87146851 

0.14234545 

0.00619717 

16 

20 

8.69 

30 

0.13733849 

2.04755912 

0.14301881 

0.00528466 

15 

35 

30.16 

60 

0.14123602 

2.29682015 

0.14348918 

0.00211253 

14 

34 

33.48 

90 

0.14325835 

2.56269162 

0.14333762 

0.00007484 

13 

41 

0.67 

120 

0.14189014 

2.77858020 

0.14279207 

0.00085559 

13 

8 

25.31 

150 

0.13862265 

2.88188764 

0.14237834 

0.00357038 

12 

59 

49.56 

180 

0.13616579 

2.83577714 

0.14242175 

0.00594246 

13 

12 

30.22 

210 

0.13652285 

2.64761705 

0.14287917 

0.00601413 

13 

42 

7.60 

240 

0.13961601 

2.37153873 

0.14339402 

0.00354993 

14 

24 

9.49 

270 

0.14278611 

2.09069583 

0.14344844 

0.00061690 

15 

12 

58.81 

300 

0.14243412 

1.88659813 

0.14284089 

0.0003759S 

15 

59 

29.51 

330 

0.13844795 

1.81033610 

0.14221981 

0.00347617 

16 

27 

51.54 

2! 

0.83698218 

14.04078286 

0.85728335 

0.01903366 

87 

39 

16.70 

2a 

0.83697639 

14.04078736 

0.85728218 

0.01903708 

87 

39 

18.34 

*  6a2  +  3a2e!  +  6[a's  -  Zkaa'ee'  cos  K]  =  14.91703924. 
t  -  Cfc'oo'  cos  ¥>'  •  e  sin  K'  =  +  0.31507212. 
I  6[a'V  -  A-oaV  cos  A']  =  +  0.25187831. 


OF   THE    ORBITS   OF   THE   FOUK   INNER   PLANETS. 


153 


ACTION 

OF  MERCURY  ON 

MARS. 

E 

log  /Co 

log  V 

logtfo 

log  N        log  /' 

logQ 

0° 

0.02698253 

0.30881667 

0.21633345 

0.0802043     9.8417843 

0.0229194 

30 

0.02453876 

0.30558607 

0.21270775 

0.0315423    9.7124165 

9.9318941 

60 

0.02139874 

0.30143126 

0.20804354 

9.9863199     9.5646986 

9.8328372 

90 

0.01882471 

0.29802213 

0.20421541 

9.9544525     9.4350567 

9.7499589 

120 

0.01734193 

0.29605696 

0.20200830 

9.9396879    9.3478315 

9.6977395 

150 

0.01696133 

0.29555239 

0.20144156 

9.9427992     9.3179221 

9.6840260 

180 

0.01752419 

0.29629857 

0.20227967 

9.9632970     9.3524331 

9.7119954 

210 

0.01887658 

0.29809086 

0.20429260 

9.9992756    9.4496853 

9.7797276 

240 

0.02088535 

0.30075154 

0.20728036 

0.0456467     9.5950386 

9.8772472 

270 

0.02335208 

0.30401640 

0.21094579 

0.0914150     9.7545935 

9.9819419 

300 

0.02583676 

0.30730232 

0.21463400 

0.1193985    9.8751689 

0.0582665 

330 

0.02741722 

0.30939106 

0.21697800 

fl.1150661    9.9072726 

0.0734518 

Si 

0.12996950 

1.81065732 

1.25057932 

0.1345541    7.5769549 

9.2010052 

22 

0.12997068 

1.81065891 

1.25058111 

0.1345506    7.5769465 

9.2010002 

E 

log  V 

/i' 

J  2 

J, 

Ft 

0° 

0.0211422 

0.148654076 

-0.10722959 

-0.015479687 

+0.015330048 

30 

9.9305073 

0.148321182 

-0.10335718 

-0.012048361 

+0.014842503 

60 

9.8323421 

0.145601872 

-0.06932482 

-0.006505466 

+0.009955397 

90 

9.7499432 

0.143412552 

-0.01386724 

-0.000353153 

+0.001978227 

120 

9.6975735 

0.143648964 

+0.04858051 

+0.004751607 

-0.006951532 

150 

9.6833581 

0.145958760 

+0.10133761 

+0.007449454 

-0.014441148 

180 

9.7108663 

0.148399366 

+0.13001805 

+0.007034457 

-0.018483798 

210 

9.7785043 

0.148942861 

+0.12663289 

+0.003626261 

-0.017996253 

240 

9.8764415 

0.146960494 

+0.09193479 

-0.001870362 

-rO.013109151 

270 

9.9817831 

0.144077345 

-0.03534552 

-0.007999542 

-0.005131979 

300 

0.0581594 

0.143355345 

-0.02759166 

-0.013127432 

+0.003797779 

330 

0.0724209 

0.145910972 

0.07970449 

-0.015871553 

+0.011287401 

2, 

9.1965248 

0.876620117* 

+0.06638728 

-0.025196883 

-0.009461257 

22 

9.1965168 

0.876623672 

+0.06638711 

-0.025196894 

-0.009461249 

*  2,(J,'  -  G")  = 

0.857586462. 

S,(J,'  -  G")  = 

0.857586592. 

154 


THE    SECULAR   VARIATIONS    OF   THE    ELEMENTS 


ACTION  OF  MERCURY  ON  MARS. 


E 

1000  X  F,         Ro            So            W,,            fl<»> 

8«* 

0° 

+0.3749543   -1.1968194   -0.10192940   -0.015991438    0.00000000 

-0.07377742 

30 

+0.9451901   -1.0712026   -0.08041942   -0.009779348   -0.38240323 

-0.05741705 

60 

+0.8478648   -0.9676664   -0.04346887   -0.004110828   -0.57689880 

-0.02992414 

90 

+0.1645035   -0.9003960   -0.00725842   -0.000153771   -0.59093068 

-0.00476371 

120 

-0.4045215   -0.8699380   +0.02266377   +0.002278066   -0.47241848 

+0.01421151 

150 

-0.2449197   -0.8749740   +0.04587694   +0.003542283   -0.26566469 

+0.02785885 

180 

+0.5450969   -0.9160775   +0.06265312   +0.003737622    0.00000000 

+0.03761136 

210 

+  1.2365773   -0.9949389   +0.07097316   +0.002525788   +0.30208776 

+0.04309857 

210 

+  1.1824210   -1.1083446   +0.06401101   -0.000941853   +0.60188486 

+0.04013864 

270 

+0.4525829   -1.2337308   +0.03097697   -0.007413721   +0.80969859 

+0.02033021 

300 

-0.2401087   -1.3158841   -0.02869644   -0.015188691   +0.78449755 

-0.01975474 

330 

-0.2482279   -1.2994033   -0.08505077   -0.018952193   +0.46386745 

-0.06072369 

Si 

+2.3057068   -6.3747320   -0.02476681   -0.030217122   +0.33706513 

-0.03149479 

Z2 

+2.3057062    6.3746456   -0.02490154   -0.030230962   +0.33665520 

-0.03161682 

D  .           —Ro  cos  v 

E 

RO  sin  v 
+  (cos  ,+cos  E)Sa  +  g  BCC'  ,+l)  sin  «S. 

-2  -Bo 
a 

0° 

-0.2038588     +1.1968194     -0.004111709     +0.015453801 

2.1703888 

30 

-0.7173754     +0.8165787     -0.007231890     +0.006582964 

1.9693580 

60 

-0.9154636     +0.3357086     -0.004043881     +0.000738872 

1.8450804 

90 

-0.8957943     -0.0984951     -0.000144266     -0.000053225 

1.8007921 

120 

-0.7408612     -0.4547261     +0.001481632     +0.001730419 

1.8210138 

150 

-0.4834776     -0.7324563     +0.000768358     +0.003457946 

1.8912954 

180 

-0.1253062     -0.9160795     -0.000961015     +0.003611962 

2.0030408 

210 

+0.3338218     -0.9514367     -0.001700736     +0.001867379 

2.1505951 

240 

+0.8448003     -0.7366597     +0.000887111     -0.000316419 

2.3200626 

270 

+  1.2254640     -0.1770222     +0.007311027     +0.001229687 

2.4674617 

300 

+  1.1635310     +0.6123066     +0.011609114     +0.009794123 

2.5090379 

330 

+0.5585553     +1.1811278     +0.005822238     +0.018035720 

2.3888945 

Si 

+0.0228415     +0.0373693     +0.004861252     +0.031012758 

12.6686243 

22 

+0.0211938     +0.0382962     +0.004824731     +0.031120471 

12.6683967 

sin  p  •  JA,<'>  +  cos  v  •  Bo(e)  =  +  0".000000008. 

OF   THE    ORBITS    OF   THE    FOUR   INNER   PLANETS.  155 

DEFERENTIAL   COEFFICIENTS. 

u  log  coeff. 

[de/dtlw    =  +       2517.5250  m'  p  3.4009738 

[dx/dt]w    =  +     46380.761  TO'  p  4.6663379 

[di/dt]w     =+         558.61256  m'  p  2.7471107 

[dn/di]oo    =+110961.28  TO'  p  5.0451714 

[dir/dt]M    =  +     46438.628  TO'  p  4.6668794 

=+1455134.1  TO'  p  6.1629030 


TO 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF 

[de/dt]^  =  +0.00033567000 

[dx/dt]00  =  +0.0061841007 

[di/dt]w  =  +0.000074481672 

[cKl/dt]w  =  +0.014794833 

[dw/dt]m  =  +0.0061918174 

[dL/dt]w  =  +0.19401785 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

[de/dt}00  +0.00036  +0.00033  +0.0003357 

e[dTr/dt]00  +0.00058  +0.00057  +0.0005775 

[di/dt]w  +0.00008  +0.00007  +0.0000745 

sin  i  [dtt/dt]w  +0.00047  +0.00048  +0.0004778 


NOTES. 

On  account  of  the  large  eccentricities  of  both  orbits  and  the  high  mutual  incli- 
nation, the  coefficients  of  the  expansion  diminish  but  slowly.  Thus  the  combined 
effect  of  all  terms  from  the  6th  to  the  llth  orders  is  l/30th  of  the  whole  with  [de/dt]o0, 
l/90th  with  [dw/dt]0o,  and  1 /200th  with  [di/dt]0<>.  Yet  all  of  the  variations  are  very 
small  on  account  of  the  smallness  of  the  mass  of  Mercury.  A  comparison  with  the 
computation  of  Mars  on  Mercury  renders  it  evident  that  a  division  into  twelve  parts 
is  sufficient  and  that  terms  of  orders  above  the  eleventh  are  wholly  inappreciable. 


156 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


ACTION  OF  VENUS  ON  MARS. 


E 

A 

B  COS  e 

B  sin  e 

1000  xg 

h 

0° 

2.41946745 

-0.9101348 

-0.4038413 

0.003995803 

1.8967820 

30 

2.47732532 

-0.5530947 

-0.8457642 

0.017525935 

1.9549495 

60 

2.63301427 

-0.0217107 

-1.0499345 

0.027008907 

2.1104659 

90 

2.85220984 

+0.5416339 

-0.9616453 

0.022657521 

2.3292431 

120 

3.07987480 

+0.9859909 

-0.6045536 

0.008954722 

2.5566759 

150 

3.25131035 

+  1.1922952 

-0.0743414 

0.000135408 

2.7282304 

180 

3.31318850 

+  1.1052676 

+0.4869209 

0.005808973 

2.7904410 

210 

3.24523275 

+0.7482272 

+0.9288436 

0.021138190 

2.7227529 

240 

3.06934808 

+0.2168441 

+  1.1330145 

0.031452377 

2.5468467 

270 

2.84005464 

-0.3465004 

+  1.0447250 

0.026741552 

2.3172324 

300 

2.62248756 

-0.7908574 

+0.6876330 

0.011585000 

2.0993294 

330 

2.47124772 

-0.9971620 

+0.1574211 

0.000607167 

1.9481304 

2, 

17.13738066* 

+0.5853997f 

+0.2492390t 

0.088805782 

14.0005408 

22 

17.13738062 

+0.5854002 

+0.2492388 

0.088805773 

14.0005387 

G 


G' 


10000  X  G" 


0° 

0.5226610 

1.8967805 

0.5226665 

0.0403053 

31 

39 

49.77 

30 

0.5223513 

1.9549433 

0.5223748 

0.1716188 

31 

7 

34.86 

60 

0.5225239 

2.1104579 

0.5225564 

0.2449047 

29 

50 

30.35 

90 

0.5229423 

2.3292377 

0.5229663 

0.1860051 

28 

17 

2.93 

120 

0.523174o 

2.5566741 

0.5231829 

0.0669458 

26 

53 

44.43 

150 

0.5230555 

2.7282304 

0.5230555 

0.0009489 

25 

58 

2.26 

180 

0.5227230 

2.7904401 

0.5227279 

0.0398246 

25 

38 

46.94 

210 

0.5224554 

2.7227494 

0.5224737 

0.1485921 

25 

58 

48.56 

240 

0.5224769 

2.5468406 

0.5225067 

0.2363523 

26 

55 

59.83 

270 

0.5227977 

2.3172260 

0.5228262 

0.2207297 

28 

21 

35.72 

300 

0.5231337 

2.0993259 

0.5231478 

0.1054853 

29 

56 

49.75 

330 

0.5230928 

1.9481302 

0.5230936 

0.0059582 

31 

12 

36.97 

2i 

3.1366929 

14.0005190 

3.1367881 

0.7338180 

170 

55 

41.07 

Si 

3.1366950 

14.0005170 

3.1367900 

0.7338528 

170 

55 

41.30 

*  6o2  +  3aV  +  6[o'2  -  Zkaa'ee'  cos  K\  =  17.13738065. 
t  6[a'V  -  kaa'c  cos  A']  =  +  0.5854002. 
\  -  6fc'aa'  cos  <p'  •  e  sin  K'  =  +  0.2492389. 


OF   THE   ORBITS   OF   THE    FOUR   INNER   PLANETS. 


157 


ACTION  OF  VENUS  ON  MAKS. 

I 

logtfo 

log  LO 

log  #„         log  N          log  P         log  Q 

0° 

0.10710693 

0.41334627 

0.33316554     0.1537292     0.0110396     0.2088768 

30 

0.10323050 

0.40834940 

0.32760279     0.1420610     9.9681345     0.1785258 

60 

0.09431348 

0.39683240 

0.31477271    0.1149515    9.8630204    0.1053425 

90 

0.08414014 

0.38365383 

0.30007678     0.0820056     9.7312249     0.0148651 

120 

0.07564158 

0.37216263 

0.28775224     0.0524079     9.609667o     9.9324836 

150 

0.07024983 

0.36559243 

0.27991042     0.0325881     9.5264184     9.8766175 

180 

0.06843912 

0.36323215 

0.27727293     0.0260737     9.4979593     9.8576733 

210 

0.07032296 

0.36568773 

0.28001691    0.0339679    9.5296356    9.8789748 

240 

0.07586487 

0.37290310 

0.28807661    0.0551373    9.6180289    9.9372082 

270 

0.08461922 

0.38427537 

0.30077024     0.0858518     9.7401822     0.0216495 

300 

0.09502686 

0.39775494 

0.31580088     0.1191144     9.8727052     0.1128332 

330 

0.10383003 

0.40912260 

0.32846370     0.1449403     9.9748268     0.1837860 

Zi 

0.51639284 

2.31668149 

1.81684091    0.5214139    8.4704208    0.154417o 

S2 

0.51639268 

2.31668136 

1.81684084     0.5214145     8.4704222     0.1544186 

E 

log  V 

Ji' 

J2             J3             f\ 

0° 

0.2088756 

0.52281610 

-0.0030557479     -0.012760938     +0.0014450479 

30 

0.1785212 

0.52261080 

-0.0058421642     -0.016264386     +0.0030263615 

60 

0.1053364 

0.52273704 

-0.0069459413     -0.015093265     +0.0037569353 

90 

0.0148609 

0.52303324 

-0.0063117580     -0.009561780     +0.0034410135 

120 

9.9324822 

0.52319006 

-0.0041023613     -0.001152260     +0.0021632475 

150 

9.8766174 

0.52307981 

-0.0007324438     +0.007882145     +0.0002660128 

180 

9.8576725 

0.52281605 

+0.0030484592     +0.015121070     -0.0017423280 

210 

9.8789719 

0.52262308 

+0.0062562314     +0.018625039     -0.0033236408 

240 

9.9372033 

0.52266847 

+0.0079298727     +0.017454980     -0.0040542164 

270 

0.0216445 

0.52292962 

+0.0074476695     +0.011924014     -0.0037382940 

300 

0.1128306 

0.52316808 

+0.0048298806     +0.003513972     -0.0024605273 

330 

0.1837858 

0.52311928 

+0.0008867096     -0.005521489     -0.0005632929 

Si 

0.1544005 

3.13739580* 

+0.0017041620     +0.007083559     -0.0008918410 

2i 

0.1544017 

3.13739582 

+0.0017042445     +0.007083543     -0.0008918399 

*  2,(J/  -  G")  =  3, 

13732242. 

2,(J,'  -  G")  =  3. 

13732243. 

158 


THE    SECULAR    VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  VENUS  ON  MARS. 

E 

100000  X  F, 

Ro         1000  X  So     1000  X  W<> 

«<»' 

S<"> 

0° 

-  2.705324 

-1.4244790    -3.460778    -20.67006 

0.0000000 

-0.002504942 

30 

-  0.803003 

-1.3866027    -6.000157    -24.54092 

-0.4949963 

-0.004283931 

60 

+  5.500746 

-1.3028006    -6.111878    -19.19609 

-0.7766973 

-0.004207440 

90 

+  9.816975 

-1.2077652    -4.678320    -  9.84176 

-0.7926572 

-0.003070385 

120 

+  7.633806 

-1.1282518    -2.631099    -  0.95528 

-0.6126954 

-0.001649853 

150 

+  0.880725 

-1.0779056    -0.461908    +  5.93579 

-0.3272800 

-0.000280495 

180 

-  3.932916 

-1.0618130    +1.648220    +10.88332 

0.0000000 

+0.000989445 

210 

-  2.161955 

-1.0812437    +3.609363    +14.08781 

+0.3282934 

+0.002191792 

240 

+  4.374565 

-1.1352348    +5.187600    +15.12313 

+0.6164876 

+0.003252928 

270 

+  9.225323 

-1.2184712    +5.773047    +12.58406 

+0.7996835 

+0.003788855 

300 

+  7.735212 

-1.3155488    +4.427365    +  4.61417 

+0.7842976 

+0.003047815 

330 

+  1.648024 

-1.3961376    +0.822269    -  8.41473 

+0.4984001 

+0.000587075 

Si 

+  18.606089 

-7.3681280    -0.940570    -10.20081 

+0.0113925 

-0.001072047 

22 

+  18.606089 

-7.3681260    -0.935706    -10.18975 

+0.0114435 

-0.001067089 

E 

„  .           —  Rt,  cos  v 
Rosmv 

+  (cos  v  +  cos  £)S0  +T-  sec2  v  +lj  sin  vS, 

Wo  sin  u 

-2^0 
a 

0° 

-0.0069216 

+  1.4244790     -0.005314674 

+0.019975127 

2.5832417 

30 

-0.7611745 

+  1.1593980     -0.018148167 

+0.016519714 

2.5492068 

60 

-1.1839523 

+0.5449664     -0.018883467 

+0.003450266 

2.4840909 

90 

-1.2020643 

-0.1220026     -0.009233404 

-0.003406539 

2.4155306 

120 

-0.9266826 

-0.6439879     -0.000621304 

-0.000725630 

2.3617340 

150 

-0.4956901 

-0.9571936     +0.001287534 

+0.005794466 

2.3299411 

180 

-0.0032964 

-1.0618130     -0.002798312 

+0.010517416 

2.3216920 

210 

+0.4917082 

-0.9631868     -0.009486007 

+0.010415471 

2.3371562 

240 

+0.9297081 

-0.6522758     -0.014244152 

+0.005080676 

2.3763511 

270 

+  1.2126214 

-0.1251906     -0.012409750 

-0.002087273 

2.4369424 

300 

+  1.1939212 

+0.5533939     -0.003526734 

-0.002975358 

2.5083988 

330 

+0.7575012 

+  1.1728178     +0.002585060 

+0.008007817 

2.5667361 

2i 

+0.0027764 

+0.1647626     -0.045388643 

+0.035322497 

14.6355085 

22 

+0.0029019 

+0.1646422     -0.045404734 

+0.035243656 

14.6355132 

sin<f>  •  j4i(<)  +  coaip  • 

Bo(c)  =  O."0000000073. 

OF   THE    ORBITS    OF   THE   FOUR    INNER    PLANETS.  159 

DIFFERENTIAL  COEFFICIENTS. 

[de/dt]w  =  +         324.6318  m'  p  2.5113911 

[dx/dfloo  =  +  201915.56      TO'  p  5.3051698 

[di/dt]oo  5236.2608  m'  n  3.7190213 

[dtt/dt]m  +  126021.28      TO'  p  5.1004439 

[dTT/dtlw  =  +  201981.28      m'  p  5.3053112 

[dL/dt]m  =  +1681713.6        m'  p  6.2257520 

FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  m'. 

[dg/dfloo  =  +0.0007954049 

[dx/dfloo  =  +0.49472856 

[di/di]oo  =  -0.012829757 

[dO/d<]oo  =  +0.30877426 

[dr/dfloo  =  +0.49488961 

[dL/dt]m  =  +4.1204933 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

loo  +0.00080  +0.00079  +0.000795 

e[drldt]ao  +0.04618  +0.04614  +0.0461574 

[difdt]m  -0.01280  -0.01284  -0.012830 

sin  i  [dfl/d<]oo  +0.00993  +0.00998  +0.009972 

[dL/dt]M  +4.117  +4.120493 

NOTES. 

The  very  close  agreement  of  the  sums  of  the  functions  near  the  beginning  of  the 
computation  is  caused  by  the  great  circularity  of  the  orbit  of  Venus.  The  discrepan- 
cies increase  however  as  the  work  proceeds  because  of  the  high  eccentricity  of  Mars  and 
the  rather  large  mutual  inclination.  All  terms  from  the  6th  to  the  llth  orders,  in- 
clusive, produce  an  effect  equal  to  l/30th  of  the  whole  in  the  very  small  coefficient 
[de/dt]oo,  and  1 /1000th  of  the  whole  in  [dttfdt}00.  Yet  it  is  evident  that  terms  of  the 
twelfth  and  higher  orders  are  wholly  inappreciable. 


160 


THE    SECULAR   VARIATIONS    OF   THE    ELEMENTS 


E 

A 

0° 

2.88085183 

30 

2.95824702 

60 

3.13279096 

90 

3.36510700 

120 

3.59664230 

150 

3.76166099 

180 

3.80855449 

210 

3.72106141 

240 

3.52632175 

270 

3.28390784 

300 

3.06247042 

330 

2.91764743 

Si 

20.00763175* 

S2 

20.00763169 

ACTION  OF  THE 

EARTH  ON  MARS. 

B  cos  f 

B  sin  « 

1000  X  g 

-0.8153552 

-1.1018750 

0.34149937 

-0.0872040 

-1.3962265 

0.54832354 

+0.6917421 

-1.2860904 

0.46523043 

+  1.3127653 

-0.8009781 

0.18045398 

+  1.6094626 

-0.0708746 

0.00141289 

+  1.5023341 

+0.7085891 

0.14122581 

+  1.0200854 

+  1.3285566 

0.49646115 

+0.2919341 

+  1.6229077 

0.74082034 

-0.4870119 

+  1.5127725 

0.64368353 

-1.1080347 

+  1.0276597 

0.29704612 

-1.4047322 

+0.2975561 

0.02490366 

-1.2976042 

-0.4819076 

0.06532098 

+0.6141908J 

+0.6800452f 

1.97319103 

+0.6141906 

+0.6800443 

1.97319077 

1.8833414 
1.9597848 
2.1332651 
2.3654511 
2.5975998 
2.7632686 
2.8103218 
2.7223893 
2.5269386 
2.2841910 
2.0633533 
1.9197321 

14.0148199 
14.0148169 


E 


I' 


G 


G' 


G' 


o 

0 

0.9972292 

1.8831367 

0.9976157 

0.00018178 

0 

46 

42 

32.02 

30 

0.9981810 

1.9594937 

0.9987522 

0.00028018 

45 

33 

35.15 

60 

0.9992447 

2.1330727 

0.9996552 

0.00021818 

43 

12 

19.69 

90 

0.9993747 

2.3653953 

0.9995068 

0.00007633 

40 

32 

44.90 

120 

0.9987612 

2.5975995 

0.9987622 

0.00000054 

38 

19 

18.09 

150 

0.9981111 

2.7632397 

0.9981913 

0.00005120 

36 

56 

40.43 

180 

0.9979514 

2.8102243 

0.9982259 

0.00017698 

36 

35 

10.16 

210 

0.9983909 

2.7222314 

0.9988211 

0.00027246 

37 

17 

7.37 

240 

0.9991020 

2.5267718 

0.9995236 

0.00025487 

38 

58 

33.58 

270 

0.9994356 

2.2840897 

0.9996671 

0.00013009 

41 

25 

15.69 

300 

0.9988359 

2.0633420 

0.9988593 

0.00001208 

44 

5 

20.00 

330 

0.9976341 

1.9196952 

0.9977050 

0.00003411 

46 

7 

50.45 

2, 

5.9911243 

14.0141470 

5.9926419 

0.00081443 

247 

53 

13.54 

22 

5.9911274 

14.0141449 

5.9926435 

0.00084437 

247 

53 

13.98 

*  Go2  +  3o2e2  +  6[a'2  -  Zkaa'ee'  cos  K]  =  20.00763172. 
t  6[a'V  -  fcaa'e  cos  A']  =  +  0.6141907. 
t  -  6fcW  cos  <p'  •  e  sin  K'  =  +  0.6800448. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


161 


ACTION 

OF  THE  EARTH  ON  MABS. 

E 

log  A'0 

log  Lo' 

lOgtfo 

log  N         log  P 

logQ 

0° 

0.25724275 

0.60257672 

0.54204483 

0.3085062     0.3612353 

0.5756272 

30 

0.24243067 

0.58425966 

0.52198206 

0.2796590    0.2795065 

0.5094350 

60 

0.21406633 

0.54897788 

0.48324184 

0.2277018     0.1185796 

0.3818936 

90 

0.18497596 

0.51250382 

0.44306081 

0.1727907    9.9374590 

0.2419337 

120 

0.16284593 

0.48455413 

0.41218099 

0.1292685    9.7846780 

0.1268772 

150 

0.15007482 

0.46834273 

0.39423491 

0.1040949     9.6895845 

0.0569033 

180 

0.14686278 

0.16425584 

0.38970669 

0.0998549     9.6665742 

0.0407933 

210 

0.15317152 

0.47227918 

0.39859494 

0.1168787    9.7192209 

0.0805051 

240 

0.16915889 

0.49254542 

0.42101792 

0.1535252     9.8408510 

0.1719333 

270 

0.19422097 

0.52412777 

0.45588060 

0.2048055     0.0114576 

0.3019483 

300 

0.22440852 

0.56187400 

0.49741669 

0.2597585     0.1924850 

0.4426015 

330 

0.24970710 

0.59326698 

0.53185208 

0.3003845    0.3271715 

0.5489966 

2, 

1.17458520 

3.15478399 

2.74560896 

1.1786151     9.9644030 

1.7397260 

22 

1.17458104 

3.15478014 

2.74560540 

1.1786131     9.9643999 

1.7397219 

E 

logF 

JV 

J2 

J, 

F, 

0° 

0.5755790 

0.99892684 

-0.018227992 

+0.028809050 

+0.018467466 

30 

0.5093633 

0.99952648 

-0.022898545 

+0.020515873 

+0.023400805 

60 

0.3818419 

0.99990323 

-0.021379458 

+0.006120062 

+0.021554920 

90 

0.2419173 

0.99967015 

-0.013761084 

-0.010518960 

+0.013424123 

120 

0.1268771 

0.99911770 

-0.001690224 

-0.024941756 

+0.001187862 

150 

0.0568937 

0.99877639 

+0.011675876 

-0.033284760 

-0.011875978 

180 

0.0407608 

0.99892204 

+0.022549333 

-0.033314577 

-0.022266656 

210 

0.0804536 

0.99942130 

+0.027739796 

-0.025024224 

-0.027199994 

240 

0.1718817 

0.99985593 

+0.025696938 

-0.010634071 

-0.025354120 

270 

0.3019193 

0.99982014 

+0.017055202 

+0.006002126 

-0.017223615 

300 

0.4425985 

0.99929729 

+0.004464507 

+0.020427743 

-0.004987052 

330 

0.5489877 

0.99880864 

-0.008398230 

+0.028776417 

+0.008076789 

2, 

1.7395389 

5.99602303* 

+  0.011413104 

-0.013533549 

-0.011397580 

Zo 

1.7395349 

5.99602310 

+0.011413015 

-0.013533528 

-0.011397570 

*Z,(Ji'  -  G")  =  5.99517860. 
22(J/  -  G")  =  5.99517873. 


162 


THE    SECULAR   VARIATIONS    OF   THE    ELEMENTS 


ACTION  OF  THE  EARTH  ON  MARS. 

E 

1000  XF, 

n                  c*                 "nr 

ft/Q                         Ofl                         If  0 

R<»> 

fl« 

0° 

-0.13911285 

-2 

.0302096   -0.026171976   +0.10810004 

0.0000000 

-0.018943511 

30 

-0.51397488 

-1 

.9019956   -0.029451014   +0.06531256 

-0.6789837 

-0.021027135 

60 

-0.65316258 

-1 

.6889696   -0.023181583   +0.01388518 

-1.0069217 

-0.015958290 

90 

-0.40694037 

-1 

.4884239   -0.012395858   -0.01871310 

-0.9768538 

-0.008135411 

120 

-0.02612649 

-1 

.3462165   -0.001540204   -0.03342035 

-0.7310608 

-0.000965798 

150 

+0.08995598 

-1 

.2702099   +0.007499119   -0.03789964 

-0.3856685 

+0.004553852 

180 

-0.20223795 

-1 

.2578222   +0.014435140   -0.03668660 

0.0000000 

+0.008665574 

210 

-0.63957154 

-1 

.3082364   +0.019136251   -0.03045222 

+0.3972143 

+0.011620521 

240 

-0.80757787 

-1 

.4237326   +0.020598-119   -0.01635704 

+0.7731558 

+0.012916412 

270 

-0.54879816 

-1 

.6023546   +0.016496265   +0.01146539 

+  1.0516264 

+0.010826511 

300 

-0.11741630 

-1 

.8174946   +0.004601712   +0.05641742 

+  1.0835451 

+0.003167836 

330 

+0.07369507 

-1 

.9931947   -0.012572900   +0.10202141 

+0.7115406 

-0.008976672 

2, 

-1.94563404 

-9 

.5644451   -0.011258492   +0.09193865 

+0.1187184 

-0.011117777 

22 

-1.94563390 

-9.5644151   -0.011288137   +0.09173440 

+0.1188753 

-0.011138334 

—  Ro  COS  V 

E 

Ro  sin  v 
+  (c,osv  +  cosE)S» 

.  (r       \  .   „      Wa  cos  u 
+  (  -  seo2  <f  +  1  j  sin  i'.S0 

W0  sin  u 

—  2  —  RQ 

a 

0° 

-0.0523440 

+2.0302096     +0.027794619 

-0.10446568 

3.6817114 

30 

-1.0803151 

+  1.5681913     +0.048299056 

-0.04396512 

3.4967331 

60 

-1.5490313 

+0.6794308     +0.013659050 

-0.00249569 

3.2204119 

90 

-1.4807799 

-0.1636143     -0.017556368 

-0.00647718 

2.9768478 

120 

-1.1074128 

-0.7656902     -0.021736270 

-0.02538610 

2.8179922 

150 

-0.5982290 

-  1  .  1  202  1  78     -  0.008220822 

-0.03699731 

2.7456152 

180 

-0.0288703 

-  1  .2578222     +0.009432839 

-0.03545319 

2.7502731 

210 

+0.5690362 

-1.1796147     +0.020504958 

-0.02251409 

2.8278111 

240 

+  1.1509410 

-0.8419068     +0.015406346 

-0.00549521 

2.9802545 

270 

+  1.5938315 

-0.1824413     -0.011306572 

-0.00190172 

3.2047092 

300 

+  1.6480563 

+0.7072282     -0.043121307 

-0.03637964 

3.4654744 

330 

+  1.0579845 

+  1.6887208     -0.031341647 

-0.09708793 

3.6643987 

2, 

+0.0613389 

+0.6114494     +0.001435277 

-0.20967551 

18.9161075 

22 

+0.0615274 

+0.6110240     +0.000378605 

-0.20894335 

18.9161151 

sin  ip  •  |Ai(>)  +  cos 

f  • 

B0<«>  =  +  0.000000102. 

OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  163 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[de/dt]w  =  +  7024.3393  TO'  p  3.8466055 
[dx/dt]w  =  +  749340.69  TO'  p  5.8746793 
[di/dt]<n  =  +  104.61082  TO'  p  2.0195766 
[dtt/dt]w  =  -  -  747594.66  TO'  n  5.8736662 
[dvldt]m  =  +  748950.76  TO'  p  5.8744532 
[dL/dt]m  =  +2175235.9  TO'  p  6.3375064 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 

[<fe/cft]oo  =  +0.021481158 

[dx/dt]w  =  +2.2915614 

[dildt]m  =  +0.00031991074 

[dQ/<ft]oo  =  -2.2862242 

[dr/dt]m  =  +2.2903688 

*dLldt]m  =  +6.6520970 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Neweomb.  Method  of  Gauss. 


[de/dt]oo 

+0.02151 

+o!o2148 

+o!o2148116 

e[dirldt]M 

+0.21276 

+0.21374 

+0.21361818 

[difdt]m 

+0.00030 

+0.00032 

+0.00031991 

sin  i  [dtt/dt]m 

-0.07391 

-0.07379 

-0.07383093 

[dL/diloo 

+6.638 

+6.6520970 

NOTES. 

As  in  all  cases  in  which  Mars  is  the  disturbed  body,  the  gradual  increase  in  the 
discrepancies  in  the  sums  of  the  functions  as  the  computation  proceeds  is  caused  prin- 
cipally by  the  large  value  of  e.  The  greatest  effect  which  is  here  produced  by  the 
inclusion  of  all  terms  from  the  fifth  to  the  eleventh  orders  occurs  with  the  coefficient 
[dirldt]QQ  and  amounts  to  0".0007.  It  is  evident  that  a  division  into  twelve  parts  is 
fully  sufficient. 


164 


THE    SECULAR   VARIATIONS    OF   THE    ELEMENTS 


ACTION  OF  JUPITER  ON  MARS. 

E 

A 

B  cos  f 

B  sin  f 

g 

h 

0° 

29.52014024 

+6.924444 

-4.477639 

1.2637451 

27.008366 

30 

29.73057269 

+  8.555992 

-0.736229 

0.0341655 

27.006182 

60 

29.83402815 

+8.090126 

+3.325863 

0.6972206 

27.007669 

90 

29.81017790 

+5.651672 

+6.620206 

2.7625134 

27.011679 

120 

29.66910846 

+  1.894010 

+8.264081 

4.3047762 

27.014415 

150 

29.44492347 

-2.175995 

+7.817013 

3.8516159 

27.013253 

180 

29.19030067 

-5.467791 

+5.398794 

1.8371936 

27.009343 

210 

28.96977017 

-7.099343 

+  1.657384 

0.1731141 

27.006389 

2-10 

28.84611897 

-6.633477 

-2.404708 

0.3644905 

27.006977 

270 

28.85987156 

-4.195019 

-5.699050 

2.0472288 

27.010168 

300 

29.01103882 

-0.437358 

-7.342925 

3.3985953 

27.012567 

330 

29.25541959 

+3.632647 

-6.895857 

2.9973538 

27.011662 

2, 

176.07073531* 

+4.369955t 

+2.7634661 

11.8660213 

162.059336 

22 

176.07073538 

+4.369955 

+2.763467 

11.8660215 

162.059333 

E 

o 

0 

30 

60 

90 

120 

150 

180 

210 

240 

270 

300 

330 

2, 


I 


G 


G' 


G" 


27.0064605 
27.0061300 
27.0066036 
27.0074648 
27.0078870 
27.0074649 
27.0066097 
27.0061336 
27.0064416 
27.0071624 
27.0075479 
27.0071916 

162.0415503 
162.0415473 

*  6a2  +  SaV  +  6[o'!  -  2fcaa'ee'  cos  A']  =  176.07073528. 
t  6[a'V  -  kaa'e  cos  K]  =  +  4.369954. 
t  -  Qk'aa'  cos  *>'  •  e  sin  K'  =  +  2.763466. 


2.448742 
2.661358 
2.763328 
2.735467 
2.591662 
2.368639 
2.117926 
1.900349 
1.77611T 
1.786671 
1.935440 
2.180725 

13.633208 
13.633209 


2.4695960 

0.0189481 

0 

17 

39' 

53.68 

2.6618860 

0.0004753 

18 

17 

56.75 

2.7737001 

0.0093077 

18 

43 

15.01 

2.7765210 

0.0368400 

18 

48 

58.35 

2.6581522 

0.0599626 

18 

28 

29.57 

2.4330416 

0.0586151 

17 

39 

45.34 

2.1522663 

0.0316074 

16 

30 

39.82 

1.9039718 

0.0033673 

15 

24 

39.18 

1.7842098 

0.0075644 

14 

55 

27.23 

1.8310752 

0.0413982 

15 

15 

16.64 

2.0032755 

0.0628165 

16 

2 

14.95 

2.2348561 

0.0496603 

16 

53 

32.61 

13.841199!) 

0.1902067 

102 

20 

0.26 

13.8413516 

0.1903562 

102 

20 

8.87 

OF   THE    ORBITS    OF   THE    FOUR   INNER   PLANETS. 


165 


ACTION  OF  JUPITEU  ON  MARS. 

E 

logtfo 

log  L0' 

log  No         log  N         log  P 

logQ 

0° 

0.03165483 

0.31498612 

0.22325507     8.3476456     5.7990872 

7.1391284 

30 

0.03402451 

0.31811150 

0.22676030    8.3623570    5.8175286 

7.1576473 

60 

0.03565117 

0.32025551 

0.22916442     8.3954324     5.8524487 

7.1929772 

90 

0.03602468 

0.32074766 

0.22971622     8.4366023     5.8931984 

7.2342427 

120 

0.0346975G 

0.31899875 

0.22775524     8.4742980     5.9283892 

7.2695995 

150 

0.03164635 

0.31497492 

0.22324250     8.4991683     5.9492925 

7.2899854 

180 

0.02757619 

0.30960108 

0.21721367    8.5057537    5.9513987 

7.2909894 

210 

0.02396325 

0.30482489 

0.21185335     8.4928484     5.9346402 

7.2731852 

240 

0.02244999 

0.30282273 

0.20960576    8.4633475    5.9029922 

7.2413643 

270 

0.02347170 

0.30417466 

0.21112344     8.4239469     5.8638333 

7.2029262 

300 

0.02598821 

0.30750252 

0.21485868     8.3844577     5.8269721 

7.1668224 

330 

0.02889111 

0.31133796 

0.21916251     8.3560126    5.8027960 

7.1428978 

?! 

0.17801795 

1.87416671 

1.32185284    0.5709348    5.2612881 

3.3008810 

2-2 

0.17802160 

1.87417159 

1.32185832     0.5709353     5.2612888 

3.3008846 

E 

logF 

li' 

Ji             J, 

F2 

0° 

7.1387522 

27.0207890 

-0.20899481      +0.30968500 

+  5.840160 

30 

7.1576379 

27.0065721 

-0.03630747      -0.02674093 

+  0.960260 

60 

7.1927926 

27.0099958 

+0.15284461      -0.36413185 

-  4.337906 

90 

7.2335125 

27.0290264 

+0.31348417      -0.61207993 

-  8.634698 

120 

7.2684108 

27.0492502 

+  0.40064241      -0.70414548 

-10.778797 

150 

7.2888221 

27.0526590 

+0.38540712      -0.61566134 

-10.195688 

180 

7.2903609 

27.0334483 

+0.26823852      -0.37034175 

-  7.041617 

210 

7.2731181 

27.0094582 

+0.08088102      -0.03392190 

-  2.161717 

240 

7.2112135 

27.0106120 

-0.12278062      +0.30345694 

+  3.136448 

270 

7.2021019 

27.0363209 

-0.28345023     +0.55139880 

+  7.433241 

300 

7.1655732 

27.0521350 

-0.35632057     +0.64347029 

+  9.577337 

330 

7.1419111 

27.0423420 

-0.32638726     +0.55499853 

+  8.991231 

2i 

3.2971031 

162.1762303* 

+0.13362954      -0.18200685 

-  3.604375 

V 

3.2971036 

162.1763786 

+0.13362735      -0.18200677 

-  3.604371 

*  S,(J,'  -  G")  =  161 

.9860236. 

2«(Ji'  -  G")  =  161 

.9860224. 

166 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


E 
0° 
30 
60 
90 
120 
150 
ISO 
210 
240 
270 
300 
330 

Si 
E 

F, 
-0.11464084 
-0.02209150 
+0.08535418 
+0.09095508 
-0.02126155 
-0.14774803 
-0.16666131 
-0.05846157 
+0.07437981 
+0.10831696 
+0.01978452 
-0.09401607 

-0.12304519 
-0.12304513 

fto  sin  v  + 
(cos  v  +  oos  E)f><, 

ACTION  OF  JUPITER  ON  MARS. 
#„        1000  X  -So       1000  X  W0     1000  X  RM     1000  X  <S("> 
0.011523729   +0.08005004   +0.4190397     0.000000   +0.05794093 
0.011964482   +0.01088811   -0.0398940   +4.271140   +0.00777378 
0.012921829   -0.07058125   -0.5615348   +7.703673   -0.04858841 
0.014185469   -0.13852471   -1.0407895   +9.309934   -0.09091387 
0.015433331   -0.17072189   -1.3081943   +8.381048   -0.10705259 
0.016299442   -0.15776374    -1.2103390   +4.948932   -0.09580229 
0.016504164   -0.10615120   -0.7376107     0.000000   -0.06372374 
0.015978601   -0.03427562   -0.0686500   -4.851517   -0.02081393 
0.014896574   +0.03689409   +0.5347724   -8.089562   +0.02313475 
0.013593080   +0.09184683   +0.8860645   -8.921148   +0.06027915 
0.012429971   +0.12131684   +0.9434385   -7.410440   +0.08351498 
0.011688531   +0.11863615   +0.7635196   -4.172629   +0.08470265 

0.083709598   -0.10919337   -0.7100892   +0.584719   -0.05477408 
0.083709605   -0.10919298   -0.7100884   +0.584712   -0.05477451 

i 

—  flo  COS  V  + 

Ir         \  .       1000  X  Wo  cosu    1000  X  JTo  sin  u        -2-R,, 

0° 

+0.000160100 

-0.011523729 

+0.10774325 

-0.4049515 

-0.020897865 

30 

+0.006498115 

-0.010046690 

-0.02950189 

+0.0268547 

-0.021996162 

60 

+0.011621455 

-0.005638033 

-0.55239000 

+0.1009292 

-0.024638457 

90 

+0.014136555 

+0.001045999 

-0.97645422 

-0.3602495 

-0.028370938 

120 

+0.012896599 

+0.008458998 

-0.85083667 

-0.9937043 

-0.032306104 

150 

+0.007784436 

+0.014315488 

-0.26253500 

-1.1815227 

-0.035231972 

180 

+0.000212302 

+0.016504164 

+0.18965406 

-0.7128120 

-0.036086942 

210 

-0.007299886 

+0.014215603 

+0.04622538 

-0.0507547 

-0.034538456 

240 

-0.012311629 

+0.008381401 

-0.50369057 

+0.1796589 

-0.031182525 

270 

-0.013542396 

+0.001084104 

-0.87379071 

-0.1469683 

-0.027186159 

300 

-0.011129593 

-0.005518211 

-0.72109467 

-0.6083574 

-0.023700620 

330 

-0.006127613 

-0.009949901 

-0.23455823 

-  0.7265980 

-0.021488839 

Si 

+0.001449234 

+0.010664590 

-2.33061460 

-2.4392371 

-0.168812513 

2,. 

+0.001449211 

+0.010664603 

-2.33061467 

-2.4392385 

-0.168812526 

sin  v>  •  Miw  +  cos  <p 

•  B0(c)  =  -  0.00000000019. 

OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  167 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[<fe/cB]oo  =  +  165.70584  m'  p  2.2193378 
[dx/dtln  =+13074.175  m'  p  4.1164143 
[di/dt]m  =  -  268.82366m'  n  2.4294675 
[dQ/dt]w  =  --  8712.2760  m'  n  3.9401316 
[dT/dfla,  =  +13069.631  TO'  p  4.1162634 
[dL/dt]00  =  -19334.282  TO'  n  4.2863281 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 

[de/dt]m  =  +  0.15813453 
[dxldt]m        +12.476799 

[di/dt]m  =  --  0.25654077 

[dQ/dflflo  =     -  8.3142000 

[dirfdt]w  =  +12.472464 

[dL/dt]w  =  -18.450874 


COMPARISON  WITH  OTHER  RESULTS. 

LeveiTier.  Newcomb.  Method  of  Gauss. 


[de/dt}oo 

+  0.15810 

+0.15818 

+  0.1581345 

e[dirfdt]m 

+  1.16323 

+  1.16372 

+  1.1632822 

[di/dfloo 

-  0.25648 

-0.25655 

-  0.2565408 

sin  i  [dttjdt]w 

-  0.26864 

-0.26850 

-  0.2684974 

[dL/dt]oo 

-18.450 

-18.450874 

NOTES. 

The  very  exact  agreement  of  the  final  sums  shows  that  for  this  case  the  expansion 
of  the  perturbing  function  is  highly  convergent.  The  greatest  effect  of  all  terms 
from  the  sixth  to  the  eleventh  orders  inclusive  occurs  with  [de/dt]00,  and  amounts  to 
but  l/100000th  of  the  whole  variation. 

This  computation  has  been  twice  effected  by  DR.  ARTHUR  B.  TURNER  from  the 
same  elements  as  are  here  employed.  His  first  computation  was  made  by  HILL'S 
first  method,  exactly  as  here,  and  was  presented  as  a  Thesis  to  the  Faculty  of  the 


168 


THE    SECULAR   VARIATIONS   OF   THE   ELEMENTS 


Graduate  School  of  the  University  of  Pennsylvania,  1902.  The  values  of  the  functions 
in  this  computation  agree  practically  throughout  with  those  here  given,  but  the  last 
two  figures  usually  differ  because  eight  place  logarithms  are  here  employed  in  certain 
parts  of  the  work. 

DR.  TURNER'S  second  computation,  (A.  N.,  3065),  was  made  according  to  the 
method  developed  by  DR.  Louis  ARNDT(SO).  The  two  papers  taken  together  are  of 
high  value  since  they  afford  a  means  of  comparing  the  labor  and  accuracy  appertaining 
to  the  two  very  different  methods.  It  is  DR.  TURNER'S  opinion  that  while  the  form- 
ulas of  ARNDT'S  method  are  presented  in  a  more  symmetric  form  yet  they  are  less 
accurate  in  application  than  those  of  DR.  HILL.  This  is  confirmed  by  the  circum- 
stance that  the  residual  from  the  equation  [da/dt}^  =  0  is  300  times  larger  with  the 
former  method  than  with  the  latter. 

DR.  TURNER'S  results  from  ARNDT'S  method,  which  agree  almost  exactly  with 
his  earlier  values  and  with  those  here  obtained,  are  as  follows: 


[de/diloo    =  +  0.1581330 
[dx/<ttjoo    =  +12.47677 
[dildt]M     =  -  0.2565480 

[dti/dt]*    =  -  8.314194 
ldr/dt]m    =  +12.47244 
[dL/dtloo   =  -18.45083 

ACTION  OF  SATURN  ON  MARS. 


E 

A 

B  sin  ( 

B  cos  < 

9 

h 

0° 

92.2326164 

-11.747198 

-  0.835808 

39.461155 

90.708401  T 

30 

93.1075498 

-  13.263728 

+  6.495252 

50.307460 

90.7105920 

60 

94.0543836 

-10.902482 

+  13.616320 

33.990063 

90.7106532 

90 

94.8268069 

-  5.296150 

+  18.619311 

8.020876 

90.7086748 

120 

95.2215463 

+  2.053054 

+20.163679 

1.205319 

90.7070348 

150 

95.1291338 

+  9.175915 

+  17.835613 

24.076867 

90.7078833 

180 

94.5669406 

+  14.163873 

+  12.258914 

57.367408 

90.7108082 

210 

93.6819094 

+  15.680401 

+  4.927852 

70.309758 

90.7131088 

240 

92.7148797 

+  13.319160 

-  2.193216 

50.728826 

90.7125327 

270 

91.9323586 

+  7.712824 

-  7.196207 

17.010933 

90.7095237 

300 

91.5477178 

+  0.363621 

-  8.740573 

0.037809 

90.7066608 

330 

91.6603255 

-  6.759241 

-  6.412507 

13.064630 

90.7063166 

^ 

560.3380844* 

+  7.250028f 

+  34.2693  16} 

182.790580 

544.256090(1 

22 

560.3380840 

+  7.250021 

+34.269314 

182.790524 

544.2560991 

*  6d2  +  SaV  +  6[a'2  - 
t  —  Gfc'aa'  cos  if'  •  e  sin 
\  6[a'V  -  kaa'e  cos  A'] 


Ikaa'ee'  cos  A']  =  +  500.3380843. 
A'  =  +  7.250024. 
=  +  34.269314. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


169 


E 


ACTION  OF  SATUKN  ON  MARS. 
G  G' 


G" 


0 
30 
60 
90 
120 
150 
180 
210 
240 
270 
300 
330 

Zi 

z, 

E 

1.2382581 
2.1110005 
3.0577732 
3.8321748 
4.2285543 
4.1352933 
3.5701752 
2.6828133 
1.7163897 
0.9368777 
0.5550998 
0.6680516 

14.3662501 
14.3662411 

log  Ko 

90.70353S2       1.5278682      0.2847474       8 
90.7043316       2.3529760      0.2357148       9 
90.7063778      3.1798913      0.1178425      10 
90.7076570      3.8561238      0.0229312      11 
90.7068811       4.2318481      0.0031400      12 
90.7048170      4.2015371      0.0631774      12 
90.7035495       3.7462608      0.1688274      11 
90.7043024       2.9540530      0.2624034      10 
90.7062481       2.0020241      0.2793497       9 
90.7074346      1.1081936      0.1692270      6 
90.7066561      0.5558543      0.0007499      4 
90.7047169       0.8409314      0.1712800       6 

544.2332508      15.2437468     0.8546569      57 
544.2332595      15.3138149      0.9247338      57 

log  LO'         log  No         log  N         log  I' 

6  50.425 
42  47.930 
59   5.400 
55  59.135 
28  42.340 
31   7.643 
58  46.967 
50  18.088 
6  39.626 
48  32.560 
29  34.176 
3  29.499 

9  38.934 
52  14.855 

logQ 

0° 

0.00656257 

0.28174175 

0.18592136 

7.5317335 

3.8955043 

5.7586694 

30 

0.00942345 

0.28554606 

0.19019807 

7.5468281 

3.9148639 

5.7782711 

60 

0.01207449 

0.28906810 

0.19415617 

7.5819826 

3.9546474 

5.8179375 

90 

0.01427105 

0.29198398 

0.19743287 

7.6263318 

4.0028084 

5.8660110 

120 

0.01561994 

0.29377356 

0.19944338 

7.6674180 

4.0458810 

5.9092061 

150 

0.01572227 

0.29390930 

0.19959586 

7.6949830 

4.0730267 

5.9366461 

180 

0.01438391 

0.29213375 

0.19760114 

7.7028806 

4.0781506 

5.9420499 

210 

0.01175197 

0.28863978 

0.19367515 

7.6895882 

4.0604630 

5.9243808 

240 

0.00828439 

0.28403181 

0.18849593 

7.6581064 

4.0241928 

5.8876296 

270 

0.00461510 

0.27914998 

0.18300711 

7.6156278 

3.9778730 

5.8401825 

300 

0.00200564 

0.27567456 

0.17909843 

7.5727522 

3.9331412 

5.7942078 

330 

0.00365091 

0.27786617 

0.18156337 

7.5415145 

3.9024823 

5.7646287 

V 

•^-1 

0.05893094 

1.71642353 

1.14471671 

5.7148732 

3.9315172 

5.1097001 

22 

0.05943475 

1.71709527 

1.14547243 

5.7148733 

3.9315172 

5.1101201 

170 


THE    SECULAR   VARIATIONS    OF   THE   ELEMENTS 


ACTION  OF  SATUKN 

ON  MARS. 

E 

logF 

/»' 

J'Z 

J3 

F2 

F, 

0° 

5.7569714 

90.9876364 

-0.64905242 

+0.2442109 

+  59.776028 

-2.2333582 

30 

5.7768667 

90.9027545 

-0.80993896 

-1.6448282 

+  67.492953 

-2.2386206 

60 

5.8172358 

90.7031717 

-0.67748734 

-3.0908929 

+55.477667 

-1.0572468 

90 

5.8658745 

90.5722076 

-0.29321392 

-3.7064154 

+26.949646 

+0.0320703 

120 

5.9091874 

90.5928360 

+0.18262899 

-3.3264146 

-10.447038 

-0.2313819 

150 

5.9362703 

90.7254544 

+0.58278840 

-2.0527628 

-46.691968 

-1.7836963 

180 

5.9410456 

90.8717164 

+0.80450796 

-0.2268316 

-72.073367 

-3.2467872 

210 

5.9228190 

90.9409399 

+0.82304924 

+  1.6620728 

-79.790278 

-3.2597888 

240 

5.8859649 

90.8876171 

+0.67261531 

+3.1078694 

-67.775016 

-1.8125358 

270 

5.8391722 

90.7221965 

+0.40924575 

+3.7232565 

-39.246977 

-0.2549584 

300 

5.7942033 

90.5707339 

+0.07363159 

+3.3433908 

-  1.850296 

+0.0267586 

330 

5.7636057 

90.8202137 

-0.30508990 

+2.0700081 

+34.394639 

-1.0495556 

Si 

5.1046084 

544.6137115* 

+0.40684409 

+0.0513320 

-36.892022 

-8.5545513 

22 

5.1046084 

544.6837666 

+0.40684061 

+0.0513310 

-36.891985 

-8.5545494 

E 

R* 

1000000  X  So 

1000  X  Wa 

1000  X  B("> 

100000  X  S("> 

1000  X[flo  sin  v+ 
(cos  v-\-  cos  E)  So] 

0° 

0.0017097618 

+  9.903284 

+0.01219946 

0.0000000 

+0.7168085 

+0.0198066 

30 

0.0017745243 

+  7.025377 

-.0.10023833 

+0.6334785 

+0.5015908 

+0.9730086 

60 

0.0019262900 

+  5.499323 

-0.20387007 

+  1.1484062 

+0.3785755 

+  1.7472855 

90 

0.0021375714 

+  5.593738 

-0.27213044 

+  1.4028896 

+0.3671175 

+2.1277319 

120 

0.0023569201 

+  3.205770 

-0.27013289 

+  1.2799221 

+0.2010205 

+  1.9382857 

150 

0.0025168546 

-  4.917236 

-0.17936963 

+0.7641820 

-0.2986000 

+  1.1679255 

180 

0.0025606191 

-  16.044422 

-0.02369074 

0.0000000 

-0.9631644 

+0.0320888 

210 

0.0024720520 

-22.805150 

+0.13539869 

-0.7505788 

-1.3848465 

-1.0986739 

240 

0.0022843230 

-19.928906 

+0.23709997 

-1.2404980 

-1.2496588 

-1.8606368 

270 

0.0020608533 

-  9.038484 

+0.25685358 

-1.3525394 

-0.5931965 

-2.0510270 

300 

0.0018652798 

+  2.997952 

+0.20817962 

-1.1120336 

+0.2063802 

-1.6842325 

330 

0.0017413378 

+  9.774776 

+0.11927113 

-0.6216313 

+0.6978895 

-0.9263634 

2, 

0.0127031938 

-  14.366999 

-0.04021465 

+0.0757967 

-0.7100385 

+0.1925973 

2» 

0.0127031934 

-14.366879 

-0.04021500 

+0.0758006 

-0.7100452 

+0.1926017 

*  2,(J,'  -  G")  = 

543.7590546. 

2Z(J,'  -  G")  = 

543.7590328. 

OF   THE    ORBITS    OF   THE    FOUR   INNER   PLANETS. 


171 


1000  X  |~-  flocosv 
L 

f 

E 

(          \      -i 

1000  X  W0  cos  u 

1000  X  W0  sin  u 

1000X-2-flo 

-  sec2  ip  +  1  1  sin  vSo 

a 

0° 

-1.7097618 

+0.00313672 

-0.01178931 

-  3.1005914 

30 

-1.4844381 

-0.07412689 

+0.06747540 

-  3.2623831 

60 

-0.8120508 

-0.20054993 

+0.03664321 

-  3.6729182 

90 

+0.2105546 

-0.25530900 

-0.09419278 

-  4.2751426 

120 

+  1.3414125 

-0.17569179 

-0.20519295 

-  4.9336670 

150 

+  2.2292256 

-0.03890712 

-0.17509913 

-  5.4402943 

180 

+2.5606191 

+0.00609135 

-0.02289425 

-  5.5988851 

210 

+  2.2161495 

-0.09117052 

+0.10010366 

-  5.3434506 

240 

+  1.3285850 

-0.22331936 

+0.07965468 

-  4.7817011 

270 

+0.2102889 

-0.25329570 

-0.04260337 

-  4.1217062 

300 

-0.8010985 

-0.15911711 

-0.13424046 

-  3.5565885 

330 

-1.4740745 

-0.03664087 

-0.11350351 

-  3.2013711 

38, 

+  1.9077055 

-0.74945012 

-0.25781908 

-25.6443513 

22 

+  1.9077060 

-0.74945010 

-0.25781973 

-25.6443479 

sin 


+  cos  <p  •  £0(c)  =  +  0.000000000031. 


DIFFERENTIAL  COEFFICIENTS. 


[deldt]m 
[d*/dt]w 
[di/dt]m 


=  +     22.022051  m' 
=  +2338.7360     TO' 
86.444970  TO' 
-  920.85894    TO' 
[dw/dt]w    =  +2338.2557     m' 
-2935.3283      TO' 


log  coeff. 
p  1.3428578 
p  3.3689812 
n  1.9367398 
n  2.9641931 
p  3.3688920 
n  3.4676567 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 


=  +0.0062891406 
=  +0.66790508 
[di/dt]M     =  -0.024687281 
[dtt/dt]M    =  -0.26298236 
[d7r/d<]oo    =  +0.66776785 
=  -0.83828212 


172  THE  SECULAR  VARIATIONS  OF  THE  ELEMENTS 

COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

[de/dt]w  +0.00627  +0.00629  +0.0062891 

e[dw/dt]w  +0.06226  +0.06226  +0.0622814 

[di/dt]m  -0.02467  -0.02468  -0.0246873 

sin  i  [dtt/dt]m  -0.00852  -0.00849  -0.0084927 

[dL/dt}w  -0.838  -0.8382821 

NOTES. 

As  in  other  similar  cases,  the  great  disagreement  of  the  sums  of  the  functions 
near  the  beginning  of  the  computation  arises  principally  from  the  term  a'2e',  but  the 
remarkably  close  agreement  of  the  final  sums  shows  that  the  expansion  of  the  per- 
turbing function  for  this  case  is  very  convergent.  The  greatest  effect  produced  by  all 
terms  from  the  sixth  to  the  eleventh  orders  inclusive  here  occurs  with  [de/dt]00  and 
amounts  to  but  l/100000th  of  the  value  of  this  coefficient. 

DR.  SAMUEL  G.  BARTON  has  effected  this  computation  from  the  same  elements 
as  are  here  employed,  making  use  of  the  formulas  developed  by  DR.  ARNDT(SO).  (A 
Thesis  presented  to  the  Faculty  of  the  Graduate  School  of  the  University  of  Pennsylvania, 
1906}.  The  results* obtained  by  him  are  as  follows: 

[de/dt]m  =  +0.0062897 

e[dw/dt]w  =  +0.0622817 

[di/dl]w  =  -0.0246873 

sin  i  [dtt/dt]M  =  -0.0084927 

[dL/dt]w  =  -0.8382857 

The  agreement  is  thus  practically  exact. 

It  is  the  conclusion  of  DR.  BARTON  that  in  spite  of  the  greater  symmetry  of  the 
equations  employed  in  the  method  of  ARNDT,  computations  effected  by  them  are 
somewhat  less  accurate  than  when  the  methods  of  HILL  are  employed.  His  residual 
arising  from  the  equation  derived  from  the  constancy  of  the  major  axis  is  eight  times 
greater  than  that  here  obtained.  (See  the  notes  to  the  computation  of  the  action  of 
Jupiter  on  Mars,  where  it  is  shown  that  DR.  TURNER  came  to  the  same  conclusion.) 


OF   THE    ORBITS   OF   THE    FOUR   INNER    PLANETS. 


173 


ACTION  OF  URANUS  ON  MARS. 


E 

A 

B  cos  t           B  sin  e             g 

h 

0° 

367.8110283 

-  8.331256      +  6.828502       37.81310 

367.49598 

45 

368.2057075 

-  5.369694      -15.243795       188.44191 

367.49649 

90 

370.1698856 

+  12.404411      -28.797937       672.53375 

367.50044 

135 

372.5672534 

+34.579222      -25.894092       543.74175 

367.50030 

180 

373.9791848 

+  48.165033       -  8.233286       54.97149 

367.49639 

225 

373.5643097 

+45.203469      +13.839014      155.31082 

367.49635 

270 

371.5799361 

+27.429370      +27.393154      608.52085 

367.49979 

315 

369.2027636 

+  5.254555      +24.489302      486.34483 

367.49949 

Zi 

1483.5400347* 

+79.667558f      -  2.809567J:     1373.83919 

1469.99259 

22 

1483.5400342 

+79.667552      -  2.809571      1373.83931 

1469.99263 

*  4a2  +  2a2c»  +  4[a'2  - 

2kaa'ee'  cos  K]  =  1483.5400348. 

t  4[a'  e'  —  kaa'e  cos  A'] 

=  +  79.667564. 

|  -  4fcW  cos  ?'  •  e  sin  A"  =  -  2.809567. 

E 

I 

G            G'           G" 

8 

0 

O 

1    II 

0 

-  0.49589 

367.49570       0.157536      0.653148       2 

41  22.746 

45 

-  0.10173 

367.49510       0.667671      0.768004       3 

34  47.145 

90 

+  1.85851 

367.49543       2.57438(1      0.710869       5 

25  12.441 

135 

+  4.25601 

367.49622       4.582935      0.322847       6 

37  54.392 

180 

+  5.67185 

367.49598       5.698515      0.026250       7 

10  10.482 

225 

+  5.25702 

367.49519       5.337362      0.079182       6 

58  20.716 

270 

+  3.26920 

367.49525       3.718991      0.445245       6 

6  25.010 

315 

+  0.89233 

367.49588       1.682505      0.786567       4 

41  47.833 

Si 

+  10.30367 

1469.98230      12.149421      1.835512      21 

23  10.679 

22 

+  10.30363 

1469.98238      12.270473      1.956599      21 

* 

52  50.086 

ACTION  OF  URANUS  ON  MARS. 

E 

log  A, 

logLo'         logATo         log  AT         logP 

logQ 

0° 

0.00071814 

0.27395868    0.17716832    6.6153321     1.7572439 

4.2264770 

45 

0.00127262 

0.27469773    0.17799967    6.6411665     1.7838479 

4.2533081 

90 

0.00292086 

0.2768938-1     0.18046982     6.7024753     1.8471866 

4.3168538 

135 

0.00437720 

0.27883326    0.18265093    6.7600912     1.9076561 

4.3771079 

180 

0.00511859 

0.27982020    0.18376077    6.7833377     1.9325907 

4.4018149 

225 

0.00484000 

0.27944936     0.18334377     6.7609874     1.9097463 

4.3789860 

270 

0.00371007 

0.27794494    0.18165196    6.7037350    1.8501248 

4.3196094 

315 

0.00219200 

0.27592287     0.17937772     6.6423516     1.7859125 

4.2555483 

2, 

0.01246766 

1.10861766    0.72305087    6.804880T    7.3871460 

7.2647550 

22 

0.01268182 

1.10890322    0.72337209    6.8048967    7.3871627 

7.2649502 

174 


THE    SECULAR   VARIATIONS   OP   THE   ELEMENTS 


ACTION  OF  UBANUS  ON  MARS. 

E 

logF 

Jl'                                                             J, 

J, 

F, 

0° 

4.2255130 

368.030126              +0.3910957 

+5.9910041 

-117.85624 

45 

4.2521750 

368.257943               -0.6874254 

+  1.3122864 

+263.09961 

90 

4.3158056 

368.150829              -1.4287110 

-4.4833753 

+497.03672 

135 

4.3766319 

367.663776              -1.2456453 

-8.0008361 

+446.91789 

180 

4.4017762 

367.403228              -0.3170169 

-7.1797383 

+  142.10202 

225 

4.3788692 

367.559090              +0.6967658 

-2.5011942 

-238.85387 

270 

4,3189529 

367.911230              +1.2249811 

+3.2942931 

-472.79092 

315 

4.2543883 

368.137688              +1.1030033 

+6.8119278 

-422.67209 

Zi 

7.2620477 

1471.495412*             -0.1296511 

-2.3778169 

+  48,19158 

2, 

7.2620644 

1471.618496              -0.1333016 

-2.3778161 

+  48.49154 

E 

F3 

1000  X  fto           1000000  X  So         1000000  X  W 

t        1000  X  -K(n) 

1000000  X  S(B> 

0° 

+  1.1362797 

0.2059026        -0.0165525        +10.07610 

0.0000000 

-0.0119808 

45 

-  5.0397791 

0.2189646        +0.3708684        +  2.31469 

+0.1087907 

+0.2605870 

90 

-  8.3838519 

0.2523915        +0.5397194          -  9.33604 

+0.1656447 

+0.3542182 

135 

-   1.8404722 

0.2886690        +0.6481209         -19.05928 

+0.1256756 

+0.3990450 

180 

+  1.6518880 

0.3050752        +0,4171470         -18.09445 

0.0000000 

+0.2504179 

225 

-  5.0490913 

0.2898966        -0.2732646        -  6.02536 

-0.1262101 

-0.1682478 

270 

-  8.9126276 

0.2535451         -0.7948804        +  6.80306 

-0.1664018 

-0.5216806 

315 

-  2.5789640 

0.2193839        -0.6003932        +12.22078 

-0.1089990 

-0.4218605 

2, 

-14.5083118 

1.0169144        +0.1454335         -10.55133 

-0.0007571 

+0.0709747 

S, 

-14.5083066 

1.0169141         +0.1453315         -10.54917 

-0.0007428 

+0.0695237 

E 

1000  X  [flo  sin  v  + 
(cos  v  +  cos  E)So] 

1000  xl-flocosv  + 
/r                  x             -,  1000000  X  W»  cos  u    1000000  X  T^o  sin  w 
(  —  sec8  <p  +  1  J  sin  vSo  1 

1000  X  -2  ^flo 

0° 

-0.00003310 

-0.20590256            +2.590763 

-  9.737343 

-0.37339681 

45 

+0.16554695 

-0.14335624            +2.077126 

•   1.021440 

-0.40904755 

90 

+0.25124093 

+0.02461951            -8.758941 

-  3.231492 

-0.50478291 

135 

+0.18971131 

+0.21763709            -8.485224 

-17.006255 

-0.61541380 

180 

-0.00083429 

+0.30507521            +4.652436 

-17.486113 

-0.66705785 

225 

-0.19106859 

+  0.21804505            +5.009009 

-  3.348852 

-0.61803079 

270 

-0.25236576 

+0.02523742            -6.708828 

•   1.128399 

-0.50709023 

315 

-0.16617611 

-0.14329578            -6.836536 

-10.129621 

-0.40983085 

Si 

-0.00199222 

+0.14902958             -8.224570 

-31.583347 

-2.05232780 

22 

-0.00198644 

+0.14903012            -8.235625 

-31.566168 

-2.05232299 

sin  if  •  \A  i  '*'  +  cos  ip  • 

BOM  =  -  0.0000000000013. 

*  2,(Ji'  -  G")  =  1469.659900. 

2i(Ji   -  G")  =  1469.661897. 

OF   THE   ORBITS   OF   THE    FOUR   INNER    PLANETS.  175 


DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[de/dt]w  =  -  o'.341 19354m'  n  9.5330008 
[dx/dt]w  =  +274.05283  TO'  p  2.4378343 
[di/dt]w  =  -  1.4239452  m'  n  0.1534933 
[dn/dfe]oo  =  -169.16430  TO'  n  2.2283087 
[dTT/dt]M  =  +273.96460  TO'  p  2.4376945 
[dL/dt]m  =  -352.43262  TO'  n  2.5470761 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  TO'. 

[de/dt]w    =  -o!o00014964631 

[dx/dt]w    =  +0.012019862 

[di/dtlw     =  -0.000062453743 

=  -0.0074194879 

=  +0.012015994 

[dL/dt]w   =  -0.015457573 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

[de/dt]m  -0.00001  -0.00001  -0.000014964631 

e[dirfdt]m  +0.00112  +0.00112  +0.0011207080 

[di/dt]m  -0.00007  -0.00006  -0.000062453743 

sin  f  [da/dtfo  -0.00023  -0.00024  -0.00023960370 

[dL/dt]w  -0.015  -0.015457573 

NOTES. 

The  greatest  error  produced  in  this  case  by  a  division  into  only  four  parts  occurs 
with  the  coefficient  [dx/dt]00  and  amounts  to  but  0". 0000001.  It  is  evident  that, 
notwithstanding  the  disagreement  of  the  sums  of  the  functions  in  the  first  part  of 
the  computation,  a  division  into  eight  parts  is  fully  sufficient. 


176 


THE    SECULAR    VARIATIONS   OF   THE    ELEMENTS 


ACTION  OF  NEPTUNE  ON  MARS. 

E 

A              B  cos  e           B  sin  e            g 

ft 

00 

906.38891911      +21.92636      -39.01745       99.36880 

904.17365 

45 

906.94215394      +47.61371      -15.38170       15.44333 

904.17339 

90 

907.26271705      +49.05172      +19.61543       25.11471 

904.17633 

135 

907.17710690      +25.39796      +45.47311      134.97137 

904.17696 

180 

906.72119273       -  9.49146      +47.04426      144.45927 

904.17395 

225 

906.14776209      -35.17881      +23.40851       35.76678 

904.17317 

270 

905.80700342      -36.61681      -11.58863       8.76590 

904.  17576 

315 

905.91280912      -12.96307      -37.44631       91.52721 

904.17627 

2i 

3626.17983231*     +24.86981t     +16.05361J     277.70868 

3616.69969 

S2 

3626.17983205      +24.86979      +16.05361      277.70869 

3616.69979 

E 

1             G            G'           G" 

e 

0 

i   tt 

0° 

2.15000       904.17353       2.20007       0.049953       2 

51  33.458 

45 

2.70349       904.17337       2.70981       0.006303       3 

8  30.703 

90 

3.02112        904.17630       3.03031       0.009166       3 

19  25.738 

135 

2.93488       904.17680       2.98505       0.050008       3 

19  16.746 

180 

2.48197        904.17377       2.54493       0.062779       3 

4  42.112 

225 

1.90932        904.17313       1.92986       0.020498       2 

39  43.146 

270 

1.56597        904.17575       1.57215       0.006167       2 

23  40.258 

315 

1.67127        904.17616       1.72990       0.058516       2 

32  56.175 

2, 

9.21906       3616.69935       9.34746       0.128065       11 

39  21.566 

S2 

9.21896       3616.69945       9.35462       0.135325       11 

40  26.770 

ACTION  OF  NEPTUNE  ON  MARS. 

E 

log  Kn         log  LO'         log  Na         log  N         log  P 

logQ 

0° 

0.00081164    0.27408332    0.17730851     6.0300471    0.3915788 

3.2510798 

45 

0.00098011    0.27430787    0.17756111     6.0560291    0.4178275 

3.2773355 

90 

0.00109700    0.27446368    0.17773639    6.1154021     0.4773508 

3.3368810 

135 

0.00109536    0.27446148    0.17773391     6.1708446    0.5327514 

3.3923012 

180 

0.00094087    0.27425556    0.17750228    6.1926623    0.5543538 

3.4138826 

225 

0.00070344    0.27393908    0.17714626    6.1704766    0.5318928 

3.3913614 

270 

0.00056913    0.27376004    0.17694484    6.1148768    0.4761252 

3.3355658 

315 

0.00064494    0.27386109    0.17705853    6.0556543    0.4169531 

3.2764317 

s, 

0.00341864     1.09656260    0.70949202    4.4529882     1.8994085 

3.3374091 

22 

0.00342385     1.09656952    0.70949981     4.4530046     1.8994248 

3.3374297 

*  4a2  +  2aV  +  4[a'2  -  Ikaa'ee'  cos  K]  =  3626.17983218. 
t  4[a'V  -  kaa'e  cos  K]  =  +  24.869793. 
t  -  4fc'aa'  cos  <p'  •  e  sin  K'  =  +  16.05361. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


177 


ACTION  OF  NEPTUNE  ON  MAES. 

E 

log  V           JY            J, 

J, 

Ft 

0° 

3.2510498      903.898094      +0.3009875 

+  15.528734 

+299.48600 

45 

3.2773317      903.965295      -0.6661748 

-  12.963466 

+  118.06523 

90 

3.3368755       902.840060      -0.3701497 

-34.790528 

-150.56206 

135 

3.3922712      902.874950      +0.9173304 

-37.166389 

-349.03766 

180 

3.4138449       903.910920      +1.0320306 

-18.699362 

-361.09725 

225 

3.3913491       904.099942      -0.1708692 

+  9.792741 

-179.67656 

270 

3.3355621       903.071970      -0.8044972 

+31.619710 

+  88.95077 

315 

3.2763966      902.764465      +0.0451253 

+33.995664 

+287.42636 

2, 

3.3373322      3613.721044*     +0.1583712 

-  6.341446 

-123.22254 

22 

3.3373486      3613.704652      +0.1254117 

-  6.341450 

-  123.22263 

E 

F,        1000  X  Ro     1000000  X  S0    1000000  X  TFo 

1000  X  fl("> 

1000000  X  S<"> 

0° 

-10.033551   0.05357158   +0.12743633   +2.7656535 

0.00000000 

+0.09223957 

45 

-  4.253985   0.05690845   -0.09526013   -2.4561209 

+0.02827449 

-0.06693360 

90 

+  2.323772   0.06500834   -0.12559190   -7.5560767 

+0.04266503 

-0.08242607 

135 

-  5.662284   0.07385617   +0.10733754   -9.1730340 

+0.03215420 

+0.06608723 

180 

-14.586465   0.07791030   +0.13821863   -4.8544449 

0.00000000 

+0.08297417 

225 

-  7.712269   0.07407227   -0.10322243   +2.4086872 

-0.03224828 

-0.06355357 

270 

+  1.985936   0.06494182   -0.14759132   +6.8479047 

-0.04262137 

-0.09686429 

315 

-  2.681773   0.05659771   +0.08359968   +6.4235154 

-0.02812010 

+0.05874051 

2, 

-20.310308   0.26143204   -0.00752826   -2.7969634 

+0.00004366 

-0.00407662 

22 

-20.310311   0.26143460   -0.00754534   -2.7969523 

+0.00006031 

-0.00565943 

E 

1  000  V  1   /?«  rrm  v 
lOOOXtflosinv        ^L               1000000 

1000000 

1000  X  —2T-R0 

+  (oos  i)+cos  E)So]    fr   2    \  .  ,  cl    XWaCosu 

X  Wo  sin  u 

a 

0° 

+0.000254873     -0.053571580     +  0.7111033 

-  2.6726716 

-0.09715013 

45 

+0.042763861     -0.037538560     -  2.2040406 

+  1.0838516 

-0.10631064 

90 

+0.064736687     +0.005812013     -  7.0890057 

-  2.6153925 

-0.13001668 

135 

+0.048623034     +0.055602449     -  4.0838509 

-  8.2138132 

-0.15745404 

180 

-0.000276437     +0.077910304     +  1.2481721 

-  4.6912376 

-0.17035369 

225 

-0.048771753     +0.055759058     -  2.0023922 

+  1.3387309 

-0.15791475 

270 

-0.064644985     +0.006352181     -  6.7530485 

-  1.1358370 

-0.12988365 

315 

-0.042545549     -0.037317274     -  3.5934362 

-  5.3243555 

-0.10573015 

2l 

+0.000070138    +0.036502918     -11.8827788 

-11.1151387 

-0.52740415 

s. 

+0.000069593     +0.036505673     -11.8837299 

-11.1155862 

-0.52740958 

sin  v  •  iAi(s>  +  cos  <p  •  BoM  =  +  0.0000000000012. 

*  2,(J/  -  G")  =  3613.592979. 

2t(Ji  -  G")  =  3613.569327. 

178  THE  SECULAR  VARIATIONS  OF  THE  ELEMENTS 

DIFFERENTIAL  COEFFICIENTS. 

log  coeff. 

[de/dt]w  =  +  0.011982  m'  p  8.0785350 
[dx/dt]M  =  +67.128215  TO'  p  1.8269051 
[di/dt]m  =  --  2.0560028m'  n  0.3130237 
[dl2/cfc]oo  =  -59.551438  m'  n  1.7748923 
[dT/dt]w  =  +67.097154  TO'  p  1.8267041 
=  -90.590942  TO'  n  1.9570848 


FINAL  VALUES  CORRESPONDING  TO  THE  ABOVE  VALUE  OF  m'. 

[de/dtlw    =  +o!()0000060823 

[dx/dt]w    =  +0.0034075236 

[dt/<ft]oo     =  -0.00010436562 

[dtt/dt]w    =  -0.0030229161 

[dir/dtlw,    =  +0.0034059472 

=  -0.0045985255 


COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

[de/dt]w    +0^00000  +0^00000  +o!o0000060823 

+0.00032  +0.00032  +0.00031766599 

-0.00011  -0.00011  -0.00010436562 

sin  i  [dtt/dt]m    -0.00009  -0.00010  -0.000097621545 

[dL/dt]w   -0.004  -0.0045985255 

NOTES. 

The  agreement  of  the  sums  of  the  functions  is  much  more  exact  throughout  than 
in  the  preceding  computation  because  e'  is  here  so  much  smaller.  The  greatest  effect 
produced  by  all  terms  from  the  fourth  to  the  seventh  orders  inclusive  is  but  0". 000001, 
and  it  is  evident  that  the  terms  of  the  eighth  and  higher  orders  are  wholly  inappreci- 
able. 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


179 


11.     THE   FINAL  VALUES  OF  THE  PERTURBATIONS. 

Combining  the  results  of  the  preceding  pages,  we  now  obtain  the  values  of  the 
perturbations  stated  in  the  following  tables.  For  comparison  with  these,  the  results 
obtained  by  LEVERRIER(V)  and  NEWCOMB(IS)  are  added,  all  of  the  results  being  reduced 
to  the  values  of  the  masses  here  adopted  and  stated  in  Article  6. 


SECULAR  PERTURBATIONS  OF  MERCURY. 
(Epoch  1850.0,  G.  M.  T.) 


Action  of  — 

r*i 

LdtJoo 

r&i 

ldt]«, 

r*-. 

Udoo 

Venus 

+0^027739414 

+2.7763615 

-0.14811133 

Earth 

+0.011476557 

+0.91448833 

-0.014040890 

Mars 

-0.000607428 

+0.02486334 

-0.000301945 

Jupiter 

+0.00319413 

+  1.5400720 

-0.049056191 

Saturn 

+0.000531095 

+0.07312263 

-0.004212776 

Uranus 

+0.000009638 

+0.00142135 

-0.000024450 

Neptune 

+0.000003320 

+0.00041901 

-0.000020027 

-0.04234673 

+5.3307482 

-0.21576761 

Action  of  — 

rda-i 

LdUoo 

f-1 

LdUoo 

1-dL-l 

L««J« 

Venus  

-1.9420214 

+2.7618772 

-3.2505323 

Earth  

-1.0037245 

+0.90700208 

-1.1935233 

Mars        

-0.01926435 

+0.02471966 

-0.03293324 

Jupiter            

-1.4795642 

+  1.5290366 

-2.2066350 

Saturn 

-0.06979662 

+0.07260205 

-0.10657405 

Uranus                      .  . 

-0.00134987 

+0.00141128 

-0.00201139 

Neptune 

-0.00044314 

+0.00041570 

-0.00060031 

-4.5161641 

+5.2970646 

-6.7928096 

COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 

[de/dt]oo 

e[dir/dt]oo 

[di/dt]oo 

sin  i  [dtt/dt]m 

[dL,'dt]w 

*  Exclusive  of  the  action  of  Uranus  and  Neptune. 

f  This  unexpectedly  large  difference  is  a  gradual  accumulation  from  all  of  the  computations  Thus,  the  residuals, 
Newcomb-Gauss,  are,  in  the  several  cases:  0".00300,  0".00151,  0".0000.3,  0".00227,  and  0".00010,  the  sum  of  which  is 
the  difference  as  found  above. 


+0.04246 

+0.04234 

+010423467 

+  1.08946 

+  1.09601 

+  1.0891018t 

-0.21586 

-0.21570 

-0.2157676 

-0.55017 

-0.55041 

-0.5505495 

-6.8190* 

-6.79281 

180 


THE    SECULAR   VARIATIONS    OF   THE   ELEMENTS 


SECULAR  PERTURBATIONS  OF  VENUS. 
(Epoch  1850.0,  G.  M.  T.) 


Action  of  — 

(-del 
LdUoo 

r*n 

L<ftJoo 

r-A-i 

UJoo 

Mercury  

-01)13012279 

—  l'l893992 

+o'6o94965089 

Earth  

—  0.04898290 

—56289701 

+0  000044940 

Mars  

—  0.001963988 

+0  74594759 

+0  0013°04280 

Jupiter  

—  0031162921 

+6  5654682 

—  0  038659982 

Saturn  

—0000675363 

+0  07935156 

—  0  0052327048 

Uranus  

+0  000005263 

+0  00278176 

-1-0  0000018240 

Neptune  

-0.000000278 

+0.00110440 

-0.0000283988 

-0.09579247 

+0.5762842 

-0.033057385 

Action  of  — 

ran 

1  dUoo 

rd*~i 

Udoo 

nun 

LdUoo 

Mercury 

+  o'()897732 

—  1  1892420 

+  0/7454252ti 

Earth  

-  7.293993 

—  56417558 

—  5  4005288 

Mars  

-  0.0473504 

+0  74586465 

—  0  09940123 

Jupiter      

-  2.7242270 

+6  5606924 

—  5  5347410 

Saturn 

-  0.0824657 

+0  07920700 

—  0  26491624 

Uranus 

-  0  0028813 

+0  00277671 

—  0  00496096 

Neptune  

—  0.0007780 

+000110304 

—  0  00148569 

-10.061922 

+0.5586460 

-10.5606087 

COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


[de/dt]M  -  0.09558 

e[dTr/dt]M  +  0.00366 

[di/diloo  -  0.03318 

sin  i  [dO/di]oo  -  0.59530 

[dL/dt]M  -10.549 


-0.09576 
+0.00392 
-0.03306 
-0.59551 


-  0.0957925 
+  0.0038229 

-  0.0330574 

-  0.5955192 
-10.5606087 


SECULAR  PERTURBATIONS  OF  THE  EARTH. 
(Epoch  1850.0,  G.  M.  T.) 


Action  of  — 

I"*] 

L<«Joo 

f-1    =F^1 
LdUoo       LrfUoo 

[dp-] 

I  dt  Joo 

Mercury  

-o'6oi  1613570 

-  OJ0999815 

+o'o025085775 

Venus      

+0.013483339 

+  3.4537341 

+0  074457966 

Mars 

-0.015723904 

+  097519611 

+0  0063443986 

Jupiter 

-0.081841849 

+  6  9652565 

-0  025114405 

Saturn  

-0.0004330571 

+  0.18725991 

-0.0054235259 

Uranus  

+0.0000172788 

+  0.00566366 

+0.0000236793 

Neptune  

-0.0000006006 

+  0.00179708 

-0.0000364953 

-0.085660150 

+  11.4789092 

+0.052760195 

OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS. 


181 


Action  of  — 

l-dj-j 
Ldt  Joo 

rdL-l 
1  dt  Joo 

Mercury  

-(X002098681 

+  0^3930935 

Venus  

-0.28462399 

+  11.232473 

Mars  

-0.007195311 

-  0.2342424 

Jupiter  

-0.16046446 

-  9.1916336 

Saturn  

-0.013188086 

-  0.4325140 

Uranus  

-0.0000784873 

-  0.0080930 

Neptune  

-0.0000432488 

-  0.0024199 

-0.46769226 

+  1.756664 

COMPARISON  WITH  OTHER  RESULTS. 

Leverrier.  Newcomb.  Method  of  Gauss. 


-0.08569 
e[dir/dt]w   +0.19254 
[dpldt]M   +0.05290 
[dq/dt]m    -0.46754 
[dL/dt]w  +1.7570* 


-0.08563 
+0.19248 
+0.05276 
-0.46768 


-0.085660 
+0.192514 
+0.052760 
-0.467692 
+  1.756664 


The  values  of  [dp/dt}oo  and  [dq/dt}QO  obtained  by  HILL  in  the  "New  Theory" 
are  given  below.  These  were  regarded  as  provisional  results  only,  and  were  derived 
from  the  numerical  values  of  the  coefficients  in  the  expansion  of  the  perturbing 
function  stated  by  LEVERRIER  in  the  Annales,  Vol.  II. 

It  may  also  be  of  interest  to  add  the  results  obtained  by  the  first  application 
ever  made  of  the  method  of  GAUSS.  This  was  a  computation  by  NICOLAI  of  the 
secular  perturbations  of  the  Earth,  the  final  values  only  being  published,  in  BODE'S 
Berliner  Jahrbuch,  1820,  pages  224-226  (Aug.  30,  1817).  These  results  are  here 
reduced  to  the  values  of  the  masses  stated  in  Article  6. 


[de/dt]w 

e[dirjdt]w 

[dp/dt]oo 

[dq/dt]w 


Hill. 


+0.0527225 
-0.4676079 


Nicolai. 

-o!()8606 
+0.19283 
+0.05182 
-0.46738 


*  Exclusive  of  the  action  of  Neptune.     If  the  value  of  this  found  above  is  included,  we  have  [dL/<i(]oo  =  1".7546  ; 
-a  less  exact  agreement. 


182 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


SECULAR  PERTURBATIONS  OF  MARS. 
(Epoch  1850.0,  G.  M.  T.) 


Action  of  — 

f-1 

Lddoo 

[41 

UtJoo 

r*-] 

UUoo 

Mercury  

+o!6o0335670 

+  0  0061841 

-l-o'bo0074482 

Venus  

+0.000795405 

+  0  4947286 

—  0  0128<>Q71V7 

Earth 

+0.021481158 

+  2  2915614 

-un  000*31  QQ11 

Jupiter 

+0.15813453 

+  12  476799 

—  0  2s)fili4077 

Saturn  

+0.006289141 

+  0  6679051 

—  0  024687281 

Uranus  

-0.000014965 

+  00120199 

—  0  000062454 

Neptune  

+0.000000608 

+  00034075 

—  0  000104366 

+0.18702155 

+  15.952606 

-0.29383023 

Action  of  — 

rdo-l 

LdiJoo 

F£l 

UUoo 

f-1 
ldt]w 

Mercury 

+  o'6l479483 

+  o'6o61918 

+  o"l940178 

Venus  

+  0.30877426 

+  0.4948896 

+  4  1904933 

Earth  

-  2.2862242 

+  2.2903688 

+  6  6520970 

Jupiter  

-  8.3142000 

+  12.472464 

—  18  450874 

Saturn     .    ... 

-  0.26298236 

+  0  6677678 

—  0  8382821 

Uranus 

-  0.00741949 

+  0  0120160 

—  0  0154576 

Neptune  

-  0.00302292 

+  0.0034059 

—  00045985 

-10.5502799 

+  15.947104 

-  8.342604 

COMPARISON  WITH  OTHER  RESULTS. 

Leverripr.  Neweomb.  Method  of  Gauss. 


+0.18703 
+  1.48645 
[«K/ctt]oo    -0.29375 
sin  i  [dtt/dfloo    -0.34099 
-8.358* 


+0.18706 
+  1.48787 
-0.29385 
-0.34066 


+0.187022 
+  1.487355 
-0.293830 
-0.340709 
-8.34260 


12.    COMPARISON  WITH  THE  RESULTS  OF  OBSERVATION. 

From  a  discussion  of  all  the  available  observations  of  the  planets  and  of  the 
Sun,  NEWCOMB  has  derived  the  most  probable  values  of  the  preceding  coefficients 
based  upon  observations  alone.  These  will  be  found  summarized  in  a  convenient 
form  on  pages  107  and  108  of  The  Elements  of  the  Four  Inner  Planets  and  the  Funda- 
mental Constants  of  Astronomy  (Supplement  to  the  American  Ephemeris  and  Nautical 
Almanac,  1897). 

*  The  value  of  [dLldt]oo  arising  from  the  action  of  Mercury  was  not  stated  by  Leverrier.  The  value  as  found  above 
has  been  added  to  his  series  of  values  in  order  to  obtain  this  sum. 


OF   THE    ORBITS   OF   THE   FOUR   INNER   PLANETS.  183 

In  order  to  compare  the  values  here  obtained  with  those  given  by  NEWCOMB 
it  is  necessary  to  notice  that  the  values  of  i  and  fl  stated  by  him  are  measured  from 
the  movable  equator  and  equinox  and  that  it  is  therefore  necessary  to  free  the  values 
of  [di/dt]00  and  [dQ/dt]00  here  given  from  the  changes  caused  by  the  motion  of  the 
ecliptic  itself.  For  this  purpose  we  first  compute  p  and  L  from  the  equations, 


[dp-]  \dq1 

p  sin  L  =  and       p  cos  L  =  Mr      , 

L  dt  Joo  L  dt  Joo 

the  secular  variations  being  those  which  belong  to  the  Earth's  orbit,  and  then  add 
the  quantities  —  p  cos  (L  —  fl)  to  the  several  determinations  of  [di/dt]Qo  and  —  p  X 
cos  i  sin  (L  —  Q)  to  those  above  given  for  sin  i  [dQ/dt]o0.  In  this  manner  the  values 
stated  in  the  following  tabulation  are  obtained. 

In  a  similar  way  it  might  appear  necessary  to  add  the  correction, 


e  tan  \i  ( sin  i    ^7      +  P  sin  (L  —  ft)  J 


to  the  values  obtained  for  e  [dwjdt]0o,  the  first  part  arising  from  the  change  due  to  the 
plane  of  the  orbit  and  the  second  from  that  produced  by  the  motion  of  the  ecliptic. 
And  in  the  case  of  the  Earth's  perihelion,  there  is  a  secular  motion  due  to  the  lack  of 
sphericity  of  the  Earth-moon  system  which  is  expressed  by  the  equation, 


dir^  mm'         (a  '\ 2. 

TT         =   I***  '  7 —        — 7\}  'I         )  > ' 

dt  Joo      2        (m  +  m')2    \a/' 


the  accented  letters  applying  to  the  moon  (Annales  de  I'Observatoire  de  Paris,  Vol.  IV, 
pages  42-46).  Employing  the  values  of  a'  and  m'  given  in  the  Astronomical  Papers 
of  the  American  Ephemeris,  Vol.  IV,  page  11,  this  correction  is  found  to  be 
+0".0157884.  But  these  last  two  corrections  need  not  here  be  applied  because  the 
values  of  the  variations  obtained  by  NEWCOMB  from  observation  have  already  been 
freed  from  their  effects. 

MERCURY. 

Newcomb.      Method  of  Gauss.         Observ.  5i  Si  t 


[de/dt]M 
e  [dv/dt]m 
[dildt}oo 

+0.0423 
+  1.0960 
+0.0676 

+0.0423 
+  1.0891 
+0.0674 

+o!t)336 
+  1.1824 
+0.0714 

-o!()087 
+0.0864 
+0.0038 

-0.0087 
+0.0933 
+0.0040 

±0.0050 
±  0  0040 

±0.0080 

sin  i  [dn/dfloo          -0.9250  -0.9234  -0.9189  +0.0061  +0.0045  ±0.0045 


184 


THE  SECULAR  VARIATIONS  OP  THE  ELEMENTS 

VENUS. 
Newcomb.       Method  of  Gauss.         Observ.  Si 


[de/dt}w 

-0.0958 

-0.0958 

-0.0946 

+0.0012 

+0.0012 

±0.0020 

e  [dv/dt]oo 

+0.0039 

+0.0038 

+0.0029 

-0.0010 

-0.0009 

±0.0020 

[dildt]oo 

+0.0034 

+0.0034 

+0.0029 

-0.0005 

-0.0005 

±0.0030 

sin  i  [dttldt]m 

-1.0600 

-1.0603 

-1.0540 

+0.0060 

+0.0063 

±0.0012 

EARTH. 

Newcomb.       Method  of  Gauss.         Observ. 


«i 


[de/dt]^ 

-0.0856 

-0.0857 

-0.0855 

+0.0001 

+0.0002 

±0.0009 

e.  [dir/dt}oo 

+0.1925 

+0.1925 

+0.1948 

+0.0023 

+  0.0023 

±0.0012 

[d*!dt}M 

-0.4677 

-0.4677 

-0.4711 

-0.0034 

-0.0034 

±0.0023 

Newcomb. 


MARS. 

Method  of  Gauss.        Observ. 


[de/dt]M 

+0.1871 

+0.1870 

+0.1900 

+0.0029 

+0.0030 

±0.0027 

e  [dw/dt]M 

+  1.4879 

+  1.4874 

+  1.4955 

+0.0076 

+0.0081 

±0.0035 

[di/dt]w 

-0.0225 

-0.0229 

-0.0226 

-0.0001 

+0.0003 

±0.0020 

sin  i  [dn/dt]oo 

-0.7263 

-0.7251 

-0.7260 

+0.0003 

-0.0009 

±0.0020 

In  the  above  tabulation  the  column  headed  61  expresses  the  residuals  from  the 
computation  of  NEWCOMB  and  that  headed  52  states  the  residuals  from  the  results 
here  obtained.  It  will  be  noticed  that  the  differences  are  very  minute  throughout, 
the  only  appreciable  improvement  arising  from  the  more  accurate  computation 
occurring  in  the  case  of  the  node  of  Mercury,  where  the  residual  is  reduced  by  its 
fourth  part. 

The  last  column  contains  the  mean  errors  of  the  observational  results.  If  we 
multiply  these  by  0.6745  to  reduce  them  to  probable  errors,  we  observe  that  in  seven 
cases  the  residuals  are  less  than  the  probable  errors;  in  five  cases  they  vary  from  one 
to  three  times  as  great  but  that  in  each  of  these  cases  where  the  divergence  is  greatest 
a  slight  change  in  the  value  of  the  masses  will  correct  the  disagreement,  and  that 
in  the  remaining  three  cases  the  difference  is  very  much  greater  than  can  be  ascribed 
to  errors  either  in  the  adopted  masses,  the  computation,  or  to  errors  in  the  obser- 
vations themselves.  These  three  cases  are: 

1.  The  motion  of  the  perihelion  of  Mercury. 

2.  The  motion  in  the  node  of  Venus. 

3.  The  motion  of  the  perihelion  of  Mars. 


OF  THE  ORBITS  OF  THE  FOUR  INNER  PLANETS.  185 

The  first  of  these  is  the  well-known  discordance.  The  second  is  well  established, 
the  discordance  between  observation  and  theory  being  nearly  eight  times  the  probable 
error,  nor  can  the  uncertainty  remaining  in  the  values  of  the  masses  account  for  more 
than  a  small  part  of  the  discrepancy.  NEWCOMB  estimates  the  mean  error  of  the 
computed  value  arising  from  this  uncertainty  as  not  more  than  ±0".0012,  so  that  with 
this  included  the  residual  is  nearly  six  times  the  probable  error.  The  third  dis- 
cordance is  the  least  of  the  three,  but  as  the  masses  of  Jupiter  and  Saturn,  the  principal 
disturbing  planets  for  this  case,  are  accurately  known,  the  uncertainty  of  the  com- 
puted results  is  almost  negligible.  NEWCOMB  estimates  the  mean  error  of  the  result 
of  computation  arising  from  the  uncertainties  in  the  masses  of  all  the  planets  as  here 
but  ±0".0004,  so  that  the  residual  remains  between  three  and  four  times  as  large  as 
the  probable  error. 


13.     COMPARISON  WITH  SEELIGER'S  HYPOTHESIS  ON  THE  CON- 
STITUTION  OF  THE   ZODIACAL  LIGHT. 

Many  hypotheses  have  been  made  for  the  purpose  of  explaining  the  discrepancies 
shown  in  the  preceding  article.  In  general,  either  the  assumption  is  made  that 
NEWTON'S  Law  of  Gravitation  is  not  strictly  accurate*  or  else  that  certain  additional 
matter  in  the  solar  system  must  be  considered  whose  attraction  has  not  hitherto 
been  allowed  for.|  The  most  recent  and  the  most  plausible  investigation  of  the 
second  kind  is  that  effected  by  SEELIGER(IO)I  (11)i  (12>  who  seeks  to  account  for  all  of 
the  appreciable  discrepancies  by  the  perturbing  effect  of  the  cloud  of  particles  known 
as  the  zodiacal  light. 

What  the  true  form  of  this  cloud  is,  and  still  more,  what  the  law  of  the  dis- 
tribution of  its  density  is,  is  very  uncertain. J  SEELIGER  assumes  that  it  can  be 
roughly  conceived  as  made  up  of  two  homogeneous  ellipsoids  of  revolution  whose 
semi  axes  have  the  values  0.24  and  1.20,  respectively.  Both  the  eccentricities  of 
these  ellipsoids  and  the  position  of  the  equator  of  the  outer  one  can  vary  within  wide 
limits  without  greatly  altering  the  values  of  the  perturbations  which  they  produce; 
the  distance  from  the  focus  to  the  center  in  each  of  them  is  arbitrarily  chosen  as 
equal  in  length  to  ten  times  the  semi  minor  axis,  and  the  equator  of  the  outer  one  is 
assumed  to  be  coincident  with  the  plane  of  the  equator  of  the  sun.  The  respective 
densities  and  also  the  two  constants  which  define  the  equatorial  plane  of  the  first 
ellipsoid  remain  as  unknown  quantities  whose  values  are  to  be  determined. 

*  See  Tisserand's  Mecanique  Celeste,  Vol.  IV,  Pages  494-542. 

fSee  Newcomb's  "  Astronomical  Constants.  .  .  ."(1",  Pages  110-120. 

t  See  the  article,  "The  Zodiacal  Light"  by  Newcomb,  in  the  Encyclopaedia  Britannica,  Vol.  XXVIII. 


186 


THE    SECULAR   VARIATIONS   OF   THE    ELEMENTS 


From  the  known  formulas  which  express  the  attraction  exerted  by  an  ellipsoid 
upon  a  point  either  wholly  within  or  without  its  surface,  the  expression  for  the  per- 
turbing force  in  any  case  can  readily  be  written,  and  from  this  the  equations  for  the 
variations  of  the  various  elements  are  derived,  each  equation  containing  five  unknown 
quantities  whose  values  are  to  be  so  determined  as  to  best  account  for  the  excess  of 
the  variations  observed  over  those  heretofore  obtained  from  the  .theory.  As  the 
ellipsoids  are  assumed  to  be  symmetrical  with  respect  to  their  axes  of  rotation, 
however,  they  will  cause  no  appreciable  perturbation  of  any  eccentricity.  The 
variation  of  the  obliquity  of  the  Earth's  orbit  was  also  not  considered  by  SEELIGER. 

There  remain  therefore  but  ten  discrepancies  to  be  represented;  namely,  those  of 
the  four  perihelia,  those  of  the  three  nodes  and  those  of  the  three  inclinations.  These 
ten  discrepancies  form  the  absolute  terms  of  ten  corresponding  equations  which  con- 
tain five  unknown  quantities.  It  is  to  be  noticed  that  in  the  "Astronomical  Constants 
..."  two  tables  of  the  theoretical  variations  are  stated  by  NEWCOMB;  the  first, 
on  page  109,  are  those  computed  from  the  values  of  the  various  masses  assumed  in 
Chapter  V;  the  second,  on  page  185,  are  those  computed  from  the  definitively  adopted 
masses.  The  latter  values  of  the  masses  are  in  closer  accordance  with  those  assumed 
in  the  present  paper  than  the  former;  the  first  values  are,  however,  the  ones  adopted 
by  SEELIGER  in  the  computation. 

The  final  results  are  as  in  the  following  table: 


Newcomb. 

Method  of 
Gauss. 

Per.  caused  by 
Zod.L't. 

Final  Residuals. 

Prob.  Errors. 

Newcomb. 

Meth.  of  Gauss. 

MERCURY. 

// 

// 

// 

// 

// 

// 

edit 

+8.64 

+9.33 

+8.49 

+0.15 

+0.84 

±0.29 

sin  i<Kl 

+0.61 

+0.45 

+0.62 

+0.01 

-0.17 

±0.54 

di 

+0.38 

+0.40 

+0.49 

-0.11 

-0.09 

±0.35 

VENUS. 

edw 

-0.10 

-0.09 

+0.05 

-0.15 

-0.14 

±0.17 

sin  idU 

+0.60 

+0.63 

+0.60 

0.00 

+0.03 

±0.22 

di 

-0.05 

-0.05 

+0.20 

-0.25 

-0.25 

±0.11 

EARTH. 

edit 

+0.23 

+0.23 

+0.09 

+0.14 

+0.14 

±0.09 

MAKS. 

edw 

+0.76 

+0.81 

+  0.56 

+0.20 

+0.25 

±0.24 

sin  idSl 

+0.03 

-0.09 

+0.21 

-0.18 

-0.30 

±0.14 

di 

-0.01 

+0.03 

-0.01 

0.00 

+0.04 

±0.15 

The  first  two  columns  of  the  table  contain  the  residuals  from  the  masses  employed 
in  the  present  paper;  the  third  column  states  the  perturbations  caused  by  the  zodiacal 
light  when  its  elements  are  derived  from  the  residuals  of  NEWCOMB 's  first  tabulation. 
As  the  five  elements  were  so  determined  as  to  represent  NEWCOMB'S  first  residuals  as 


OF   THE    ORBITS   OF   THE    FOUR   INNER   PLANETS.  187 

accurately  as  possible,  their  agreement  with  these  is  naturally  more  exact  than  with 
the  values  here  stated.  Thus  the  first  agreement  for  the  motion  of  Mercury's  peri- 
helion is  exact  while  here  the  discrepancy  is  considerable.  On  the  other  hand,  the 
greatest  discrepancy  when  the  results  are  compared  with  the  first  tabulation,  and 
which  occurs  in  the  motion  of  the  node  of  Mars,  is  slightly  lessened  when  the  new 
masses  are  employed. 

As  the  five  elements  were  determined  to  represent  the  ten  residuals  of  NEWCOMB'S 
computation  as  accurately  as  possible,  the  numbers  of  the  fourth  column  are,  as 
might  have  been  expected,  generally  smaller  than  those  of  the  fifth.  It  may  justly 
be  inferred,  however,  that  SEELIGER'S  hypothesis  is  capable  of  greatly  reducing  those 
discrepancies  whose  values  are  sufficiently  large  to  establish  their  reality,  without 
at  the  same  time  unduly  increasing  any  of  the  smaller  ones. 

The  last  column  contains  NEWCOMB'S  estimate  of  the  total  probable  errors 
arising  both  from  the  errors  of  observation  and  from  the  uncertainties  in  the  values 
of  the  adopted  masses. 

The  elements  of  the  zodiacal  light  derived  by  SEELIGER  are  as  follows: 

Density  of  inner  ellipsoid  =  2.52  X  10~n  times  the  Sun's  density. 

Density  of  outer  ellipsoid  =  0.0026  X  10~n  times  the  Sun's  density. 

Total  mass  =  35000  X  10"11  times  the  Sun's  mass. 

Inclination  of  equator  of  I  =  6°. 95 

Longitude  of  node  of  I  =  40°.03. 

The  unit  of  time  throughout  this  article  is  the  Julian  Century. 


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